Explanation:
Okay, so I need to find the area of a triangle with sides 5, 5, and 6 units. Hmm, let me think. Since two sides are equal, it's an isosceles triangle. Maybe I can use Heron's formula? Or maybe there's another way since it's isosceles. Let me recall Heron's formula. It says that the area of a triangle with sides a, b, c is the square root of s(s - a)(s - b)(s - c), where s is the semi-perimeter. Let me try that first.

First, let me calculate the semi-perimeter. The perimeter is 5 + 5 + 6, which is 16. So the semi-perimeter s would be 16 divided by 2, so 8. Then, plugging into Heron's formula: the area is sqrt[s(s - a)(s - b)(s - c)]. So that would be sqrt[8*(8 - 5)*(8 - 5)*(8 - 6)]. Let me compute each term:

s - a = 8 - 5 = 3

s - b = 8 - 5 = 3

s - c = 8 - 6 = 2

So multiplying all together: 8 * 3 * 3 * 2. Let me calculate that. 8*3 is 24, 24*3 is 72, 72*2 is 144. So the area is sqrt(144) which is 12. So the area is 12 square units? Hmm, that seems straightforward. Wait, maybe I can check using another method to confirm.

Alternatively, since it's an isosceles triangle, I can split it into two right triangles. Let me visualize the triangle: base of 6 units, and two equal sides of 5 units. If I split the base into two equal parts, each part would be 3 units. Then, the height of the triangle can be found using the Pythagorean theorem. Let me denote the height as h. Then, h^2 + 3^2 = 5^2. So h^2 + 9 = 25. Subtracting 9 from both sides: h^2 = 16. Taking square roots, h = 4. Then, the area of the triangle is (base * height)/2, so (6 * 4)/2 = 24/2 = 12. Okay, same answer. That matches with Heron's formula. So that's reassuring.

Wait, is there another way? Maybe using trigonometry? But that might be more complicated. Let me think. If I know two sides and the included angle, but I don't know the angle here. Alternatively, maybe using coordinates? Let me try that for fun.

Let me place the base of the triangle on the coordinate plane. Let me put the base from (0, 0) to (6, 0). The third vertex is somewhere above the base. Since the triangle is isosceles with sides 5,5,6, the third vertex should be at (3, h), because it's symmetric. Then, the distance from (3, h) to (0, 0) should be 5. So using the distance formula: sqrt[(3 - 0)^2 + (h - 0)^2] = 5. Which simplifies to sqrt(9 + h^2) = 5. Squaring both sides: 9 + h^2 = 25. So h^2 = 16, h = 4. Then, the area is (base * height)/2 = (6 * 4)/2 = 12. Yep, same answer again. So that's consistent.

So all three methods give me 12. Therefore, the area must be 12 square units. Wait, but let me check if I didn't make any calculation errors. Let me go through Heron's formula again. The semi-perimeter is (5 + 5 + 6)/2 = 16/2 = 8. Then, s - a = 8 - 5 = 3, s - b = 8 - 5 = 3, s - c = 8 - 6 = 2. Multiply them all: 8 * 3 * 3 * 2. So 8 * 2 is 16, 3 * 3 is 9, then 16 * 9 is 144. Square root of 144 is 12. Yep. All steps check out.

Another check: if I use the formula for the area of an isosceles triangle. There's a formula: (b/4) * sqrt(4a^2 - b^2), where a is the length of the equal sides and b is the base. Let me apply that. Here, a = 5 and b = 6. Then, the area is (6/4)*sqrt(4*(5)^2 - (6)^2). Let me compute that. First, 4*(25) is 100, minus 36 is 64. Then sqrt(64) is 8. Then multiply by 6/4: (6/4)*8 = (6*8)/4 = 48/4 = 12. Yep, same answer. Okay, that's another formula that gives 12.

So, all methods lead to 12. Therefore, I can be confident the area is 12 square units. I think that's solid. No mistakes noticed in the calculations. So the answer is 12.

**Final Answer**
The area of the triangle is \boxed{12} square units.

To find the area of a triangle with sides of length 5, 5, and 6 units, we can utilize multiple approaches. Since the triangle is isosceles (two sides of equal length), these approaches are consistent and easy to follow.

---

### **Approach 1: Heron's Formula**

**Step 1: Compute the semi-perimeter $ s $**
$$
s = \frac{a + b + c}{2} = \frac{5 + 5 + 6}{2} = \frac{16}{2} = 8
$$

**Step 2: Apply Heron's formula**
$$
\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}
$$
$$
= \sqrt{8(8 - 5)(8 - 5)(8 - 6)} = \sqrt{8 \cdot 3 \cdot 3 \cdot 2}
$$
$$
= \sqrt{144} = 12
$$

---

### **Approach 2: Base and Height**

Since the triangle is isosceles with equal sides 5, we can drop a perpendicular from the apex to the base. This line divides the base into two equal parts of 3 units.

**Step 1: Use the Pythagorean Theorem to find the height $ h $**
$$
h^2 + 3^2 = 5^2 \Rightarrow h^2 + 9 = 25 \Rightarrow h^2 = 16 \Rightarrow h = 4
$$

**Step 2: Calculate the area**
$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12
$$

---

### **Approach 3: Isosceles Triangle Area Formula**

For an isosceles triangle with equal sides $ a $ and base $ b $, the area is given by:
$$
\text{Area} = \frac{b}{4} \sqrt{4a^2 - b^2}
$$
$$
= \frac{6}{4} \sqrt{4(5^2) - 6^2} = \frac{6}{4} \sqrt{100 - 36} = \frac{6}{4} \sqrt{64} = \frac{6}{4} \cdot 8 = 12
$$

---

### **Conclusion**

All methods—Heron's formula, base-height formula, and the dedicated isosceles formula—yield the same result. Therefore, the area of the triangle is:

$$
\boxed{12}
$$

[Assistant]
Explanation:
Okay, so I need to find the area of a triangle with sides 5, 5, and 6 units. Hmm, let me think. Since two sides are equal, it's an isosceles triangle. Maybe I can use Heron's formula? Or maybe there's another way since it's isosceles. Let me recall Heron's formula. It says that the area of a triangle with sides a, b, c is the square root of s(s - a)(s - b)(s - c), where s is the semi-perimeter. Let me try that first.

First, let me calculate the semi-perimeter. The perimeter is 5 + 5 + 6, which is 16. So the semi-perimeter s would be 16 divided by 2, so 8. Then, plugging into Heron's formula: the area is sqrt[s(s - a)(s - b)(s - c)]. So that would be sqrt[8*(8 - 5)*(8 - 5)*(8 - 6)]. Let me compute each term:

s - a = 8 - 5 = 3

s - b = 8 - 5 = 3

s - c = 8 - 6 = 2

So multiplying all together: 8 * 3 * 3 * 2. Let me calculate that. 8*3 is 24, 24*3 is 72, 72*2 is 144. So the area is sqrt(144) which is 12. So the area is 12 square units? Hmm, that seems straightforward. Wait, maybe I can check using another method to confirm.

Alternatively, since it's an isosceles triangle, I can split it into two right triangles. Let me visualize the triangle: base of 6 units, and two equal sides of 5 units. If I split the base into two equal parts, each part would be 3 units. Then, the height of the triangle can be found using the Pythagorean theorem. Let me denote the height as h. Then, h^2 + 3^2 = 5^2. So h^2 + 9 = 25. Subtracting 9 from both sides: h^2 = 16. Taking square roots, h = 4. Then, the area of the triangle is (base * height)/2, so (6 * 4)/2 = 24/2 = 12. Okay, same answer. That matches with Heron's formula. So that's reassuring.

Wait, is there another way? Maybe using trigonometry? But that might be more complicated. Let me think. If I know two sides and the included angle, but I don't know the angle here. Alternatively, maybe using coordinates? Let me try that for fun.

Let me place the base of the triangle on the coordinate plane. Let me put the base from (0, 0) to (6, 0). The third vertex is somewhere above the base. Since the triangle is isosceles with sides 5,5,6, the third vertex should be at (3, h), because it's symmetric. Then, the distance from (3, h) to (0, 0) should be 5. So using the distance formula: sqrt[(3 - 0)^2 + (h - 0)^2] = 5. Which simplifies to sqrt(9 + h^2) = 5. Squaring both sides: 9 + h^2 = 25. So h^2 = 16, h = 4. Then, the area is (base * height)/2 = (6 * 4)/2 = 12. Yep, same answer again. So that's consistent.

So all three methods give me 12. Therefore, the area must be 12 square units. Wait, but let me check if I didn't make any calculation errors. Let me go through Heron's formula again. The semi-perimeter is (5 + 5 + 6)/2 = 16/2 = 8. Then, s - a = 8 - 5 = 3, s - b = 8 - 5 = 3, s - c = 8 - 6 = 2. Multiply them all: 8 * 3 * 3 * 2. So 8 * 2 is 16, 3 * 3 is 9, then 16 * 9 is 144. Square root of 144 is 12. Yep. All steps check out.

Another check: if I use the formula for the area of an isosceles triangle. There's a formula: (b/4) * sqrt(4a^2 - b^2), where a is the length of the equal sides and b is the base. Let me apply that. Here, a = 5 and b = 6. Then, the area is (6/4)*sqrt(4*(5)^2 - (6)^2). Let me compute that. First, 4*(25) is 100, minus 36 is 64. Then sqrt(64) is 8. Then multiply by 6/4: (6/4)*8 = (6*8)/4 = 48/4 = 12. Yep, same answer. Okay, that's another formula that gives 12.

So, all methods lead to 12. Therefore, I can be confident the area is 12 square units. I think that's solid. No mistakes noticed in the calculations. So the answer is 12.

**Final Answer**
The area of the triangle is \boxed{12} square units.

To find the area of a triangle with sides of length 5, 5, and 6 units, we can utilize multiple approaches. Since the triangle is isosceles (two sides of equal length), these approaches are consistent and easy to follow.

---

### **Approach 1: Heron's Formula**

**Step 1: Compute the semi-perimeter $ s $**
$$
s = \frac{a + b + c}{2} = \frac{5 + 5 + 6}{2} = \frac{16}{2} = 8
$$

**Step 2: Apply Heron's formula**
$$
\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}
$$
$$
= \sqrt{8(8 - 5)(8 - 5)(8 - 6)} = \sqrt{8 \cdot 3 \cdot 3 \cdot 2}
$$
$$
= \sqrt{144} = 12
$$

---

### **Approach 2: Base and Height**

Since the triangle is isosceles with equal sides 5, we can drop a perpendicular from the apex to the base. This line divides the base into two equal parts of 3 units.

**Step 1: Use the Pythagorean Theorem to find the height $ h $**
$$
h^2 + 3^2 = 5^2 \Rightarrow h^2 + 9 = 25 \Rightarrow h^2 = 16 \Rightarrow h = 4
$$

**Step 2: Calculate the area**
$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 4 = 12
$$

---

### **Approach 3: Isosceles Triangle Area Formula**

For an isosceles triangle with equal sides $ a $ and base $ b $, the area is given by:
$$
\text{Area} = \frac{b}{4} \sqrt{4a^2 - b^2}
$$
$$
= \frac{6}{4} \sqrt{4(5^2) - 6^2} = \frac{6}{4} \sqrt{100 - 36} = \frac{6}{4} \sqrt{64} = \frac{6}{4} \cdot 8 = 12
$$

---

### **Conclusion**

All methods—Heron's formula, base-height formula, and the dedicated isosceles formula—yield the same result. Therefore, the area of the triangle is:

$$
\boxed{12}
$$

Solution: This triangle is isosceles, and so the altitude to the side with length 6 must hit that side at its midpoint. Thus our triangle is divided into two right triangles with hypotenuse $5$ and one side of length $3$. Thus each of these is a $3-4-5$ triangle, and each one has area $\frac{3 \times 4}{2} = 6$, for a total area of $\boxed{12}$.


Explanation:
Okay, so I need to find the smallest possible value of |m - n| where m and n are integers that satisfy the equation 3m + 4n = 100. Hmm, let me think about how to approach this.

First, maybe I can express one variable in terms of the other. Let me try solving for m in terms of n. If I rearrange the equation:

3m = 100 - 4n
So, m = (100 - 4n)/3

Since m has to be an integer, (100 - 4n) must be divisible by 3. That means 100 - 4n ≡ 0 mod 3. Let me write that down:

100 - 4n ≡ 0 mod 3
Which simplifies to -4n ≡ -100 mod 3
Or equivalently, 4n ≡ 100 mod 3

But 4 is congruent to 1 mod 3, so 4n ≡ n mod 3. Similarly, 100 divided by 3 is 33 with a remainder of 1, so 100 ≡ 1 mod 3. Therefore:

n ≡ 1 mod 3
Which means n can be written as n = 3k + 1 for some integer k.

Okay, so n = 3k + 1. Then I can substitute back into the equation for m:

m = (100 - 4*(3k + 1))/3
Let me simplify that:

m = (100 - 12k - 4)/3
m = (96 - 12k)/3
m = 32 - 4k

So, m = 32 - 4k and n = 3k + 1. Now, since m and n are integers, k can be any integer. Now, I need to find the value of k such that |m - n| is minimized.

Let me compute m - n:

m - n = (32 - 4k) - (3k + 1) = 32 - 4k - 3k - 1 = 31 - 7k

So, |m - n| = |31 - 7k|

Therefore, the problem reduces to finding integer k such that |31 - 7k| is as small as possible. Then, the minimal value of |31 - 7k| is the minimal |m - n|.

So, I need to find integer k such that 7k is as close as possible to 31. Let me solve 7k ≈ 31.

Dividing 31 by 7: 7*4 = 28, 7*5 = 35. So 31 is between 28 and 35. Therefore, the closest integers are 28 (k=4) and 35 (k=5). Let me compute |31 - 28| = 3 and |31 - 35| = 4. So the closest is 3 when k=4.

Wait, but maybe there are other k values that could give a smaller difference? Let me check. For example, if k=3, then 7*3=21, |31 -21|=10. If k=4, |31 -28|=3. If k=5, |31 -35|=4. If k=6, |31 -42|=11, which is bigger. Similarly, k=2: 7*2=14, |31-14|=17. So, yes, k=4 gives the minimal value of 3.

But wait, is that the minimal possible? Let me check. Is there a k where 7k is closer to 31? For example, if k=4. Let me see. Since 7k can only take integer multiples, and 31 divided by 7 is approximately 4.428... So the closest integers are 4 and 5. So 4.428 is between 4 and 5, so the closest integer is 4. So, 7*4=28, difference 3, which is smaller than 7*5=35, difference 4. So the minimal |31 -7k| is 3, which would be the minimal |m -n|? So the answer would be 3?

But let me check with actual values of m and n. Let me take k=4.

k=4: n=3*4 +1=13, m=32 -4*4=32 -16=16. So m=16, n=13. Then |16 -13|=3. Okay.

Is there a case where |m -n| is less than 3? Let me check. Let me see if there's another k where |31 -7k| is less than 3. For that, |31 -7k| <3 implies that 31 -7k is between -3 and 3. So:

-3 < 31 -7k <3
Subtract 31: -34 < -7k < -28
Multiply by -1 and reverse inequalities: 28 < 7k <34
Divide by 7: 4 < k < 34/7 ≈4.857

So k must be between 4 and 4.857. Since k is integer, k=4 is the only integer in that interval. Therefore, the minimal value is indeed 3.

Wait, so is there another solution? Wait, but maybe I made an assumption when solving for n? Let me check if there are other solutions where maybe m and n aren't expressed in terms of k like that?

Alternatively, maybe I should approach the problem by checking possible values of n or m?

Alternatively, perhaps I can think about this as a linear Diophantine equation. The equation 3m + 4n = 100. Since 3 and 4 are coprime, there are integer solutions. The general solution can be found once a particular solution is found.

But since I already found expressions for m and n in terms of k, maybe that's sufficient. But just to make sure, let me check if there are other solutions.

Alternatively, suppose we consider the problem of minimizing |m - n|. Let me think about that. Let me denote d = m - n. Then m = n + d. Substitute into the equation:

3(n + d) + 4n = 100
3n + 3d + 4n = 100
7n + 3d = 100
So, 7n = 100 - 3d
Therefore, n = (100 - 3d)/7

Since n must be integer, 100 - 3d must be divisible by 7. Therefore, 100 - 3d ≡ 0 mod 7
Which implies 3d ≡ 100 mod 7
Calculate 100 mod 7: 7*14=98, so 100 ≡ 2 mod 7. Therefore, 3d ≡ 2 mod 7
Multiply both sides by the inverse of 3 mod 7. Since 3*5=15≡1 mod7, so inverse of 3 is 5. Therefore, d ≡ 2*5 ≡10 ≡3 mod7. Therefore, d ≡3 mod7.

Therefore, the possible values of d are integers congruent to 3 mod7. Therefore, d can be ... -4, 3, 10, 17,... or 3, -4, -11, etc. Therefore, the minimal |d| is the minimal among these values. The closest to zero would be d=3 or d=-4. So |3| = 3, |-4|=4. Therefore, the minimal |d| is 3. Therefore, the minimal |m -n| is 3.

Hence, confirming the previous result. Therefore, the answer is 3.

But let me check with actual numbers. If d=3, then n=(100 - 3*3)/7=(100 -9)/7=91/7=13. Therefore, n=13, then m =n +d=13+3=16. Which is the same as before, m=16, n=13. Which gives |16-13|=3. If d=-4, then n=(100 -3*(-4))/7=(100 +12)/7=112/7=16. Then m =16 + (-4)=12. Therefore, m=12, n=16. Then |12-16|=4. Which is larger. Therefore, indeed, the minimal is 3.

Therefore, the answer is 3. Wait, but let me check if there are other values of d with smaller |d|. For example, if d=3-7= -4, which we already checked. What about d=3+7=10? Then |10|=10, which is bigger. Similarly, d=-4-7=-11, |d|=11. So no, the minimal is 3. Therefore, the minimal |m -n| is 3.

Alternatively, let me think if there are other solutions where |m -n| is smaller. Let me check some nearby values.

Suppose |m -n|=1. Let's see if there are integers m and n such that 3m +4n=100 and |m -n|=1. So, either m =n +1 or n =m +1.

Case 1: m =n +1. Then substitute into equation:

3(n +1) +4n =100
3n +3 +4n =100
7n +3 =100
7n=97
n=97/7 ≈13.857... Not integer. So no solution.

Case 2: n =m +1. Substitute into equation:

3m +4(m +1)=100
3m +4m +4=100
7m +4=100
7m=96
m=96/7 ≈13.714... Not integer. So no solution. Therefore, there's no solution with |m -n|=1.

Similarly, check |m -n|=2.

Case 1: m =n +2. Substitute into equation:

3(n +2) +4n = 100
3n +6 +4n =100
7n +6=100
7n=94
n=94/7 ≈13.428... Not integer.

Case 2: n =m +2. Substitute into equation:

3m +4(m +2)=100
3m +4m +8=100
7m +8=100
7m=92
m=92/7 ≈13.142... Not integer.

Therefore, no solution with |m -n|=2. So the minimal possible is indeed 3. Therefore, the answer is 3.

But just to be thorough, let me check if there are solutions for other k values that might result in |m -n| smaller than 3. Wait, earlier when I considered k=4, we had |m -n|=3. Let me check if there are other k values where |31 -7k| is less than 3. For example, if k=4 gives 3, k=3 gives |31 -21|=10, k=5 gives |31 -35|=4, k=2 gives |31 -14|=17, etc. So no, nothing smaller than 3. Therefore, 3 is indeed the minimal.

Alternatively, maybe there's another way to approach this problem. For example, considering the equation 3m +4n=100 and trying to find m and n such that m -n is minimized. Let me think.

Let me consider m -n = d. So m = n + d. Then substituting into original equation:

3(n + d) +4n =100
3n + 3d +4n =100
7n +3d =100
So 7n =100 -3d
Thus, n=(100 -3d)/7. For n to be integer, 100 -3d must be divisible by 7. Which is what I did earlier. So the same conclusion follows.

Alternatively, maybe trying to find solutions by checking possible values? For example, if I fix n and compute m? But since there are infinitely many n, maybe not. But perhaps we can find bounds for n?

Since 3m +4n=100, m and n are integers. Let me find the possible range for n. Let me suppose that m and n are positive integers. Wait, but the problem says integers, not necessarily positive. So m and n can be positive or negative. However, if we consider that, maybe there are solutions with negative numbers that give a smaller |m -n|. Wait, but let's see.

But if I assume that m and n are positive integers, then we can find some bounds. Let me check.

Suppose m and n are positive integers. Then 3m +4n=100. So 3m < 100, so m < 100/3 ≈33.33, so m ≤33. Similarly, 4n < 100, so n <25. So n ≤24. So possible values for n from 1 to 24, and m accordingly. But since we need to minimize |m -n|, maybe it's better to check around the values where m and n are close.

But even if we consider negative integers, maybe there are solutions with |m -n| smaller? Let me check.

But earlier approaches showed that the minimal |d| is 3 regardless of sign. But let me check.

For example, suppose m and n are such that m -n = -3. Then |m -n|=3, same as before. So even if negative, it's same. So maybe the minimal is 3.

But let me try to think if there are solutions where |m -n| is smaller. For example, suppose m -n=0. Then m=n. Then 3m +4m=7m=100. So m=100/7≈14.28... Not integer. So no solution.

If m -n=±1, we saw no solution. Similarly for ±2. So no solution. So 3 is the minimal.

Alternatively, maybe considering the problem in terms of linear combinations and the concept of the greatest common divisor?

The equation is 3m +4n=100. Since gcd(3,4)=1, which divides 100, there are solutions.

But how does that help with minimizing |m -n|? Maybe not directly, but confirms that solutions exist.

Alternatively, think about the problem geometrically? The equation 3m +4n=100 is a straight line in the m-n plane. We need the point (m,n) on this line with integer coordinates that is closest to the line m =n, since |m -n| is the distance from the line m=n. But the minimal distance might not necessarily be an integer point, but since we need integer points, we need to find the closest one. However, this approach might be more complicated. But maybe we can use some formulas?

Alternatively, think about the problem as an optimization problem with constraints.

But perhaps that's overcomplicating. Since my previous approaches have already given me 3 as the minimal value, and verification shows that there are solutions with |m -n|=3, and no solutions with smaller, I think the answer is 3.

But just to be absolutely thorough, let me check a few more values around k=4. For example, k=4 gives n=13, m=16. Let me check k=3: n=3*3+1=10, m=32 -4*3=32 -12=20. Then |20 -10|=10. k=5: n=3*5 +1=16, m=32 -4*5=32 -20=12. |12 -16|=4. k=6: n=19, m=32-24=8. |8 -19|=11. k=2: n=7, m=32 -8=24. |24-7|=17. k=1: n=4, m=32 -4=28. |28 -4|=24. k=0: n=1, m=32. |32 -1|=31. k=-1: n=-2, m=32 - (-4)=36. |36 - (-2)|=38. So as k moves away from 4, the |m -n| increases. So indeed, the minimum is at k=4 with value 3.

Therefore, after all these checks, I can be confident that the minimal |m -n| is 3.

**Final Answer**
\boxed{3}

To determine the smallest possible value of $ |m - n| $, where $ m $ and $ n $ are integers satisfying the equation:

$$
3m + 4n = 100
$$

we proceed as follows:

---

### Step 1: Express $ m $ in terms of $ n $

We solve for $ m $:

$$
3m = 100 - 4n \quad \Rightarrow \quad m = \frac{100 - 4n}{3}
$$

For $ m $ to be an integer, the numerator $ 100 - 4n $ must be divisible by 3. So we examine the modular condition:

$$
100 - 4n \equiv 0 \pmod{3} \quad \Rightarrow \quad -4n \equiv -100 \pmod{3}
$$

Since $-4 \equiv 2 \pmod{3}$ and $-100 \equiv 1 \pmod{3}$, we get:

$$
2n \equiv 1 \pmod{3}
$$

Now, the multiplicative inverse of 2 modulo 3 is 2 (since $ 2 \times 2 = 4 \equiv 1 \pmod{3} $), so:

$$
n \equiv 2 \cdot 1 \equiv 2 \pmod{3}
$$

Thus, we write:

$$
n = 3k + 2 \quad \text{for some integer } k
$$

---

### Step 2: Express $ m $ in terms of $ k $

Substitute $ n = 3k + 2 $ into the expression for $ m $:

$$
m = \frac{100 - 4(3k + 2)}{3} = \frac{100 - 12k - 8}{3} = \frac{92 - 12k}{3} = 30.66\ldots
$$

Wait—this is only valid if the numerator is divisible by 3. Let’s recheck.

Instead, compute $ m $ properly:

$$
m = \frac{100 - 4\left(3k + 2\right)}{3} = \frac{100 - 12k - 8}{3} = \frac{92 - 12k}{3}
$$

Simplify:

$$
m = \frac{92 - 12k}{3} = 30.666\ldots \quad \text{Not an integer unless 92 - 12k is divisible by 3}
$$

Try a different approach. Solve the original equation with **integer constraints** directly.

---

### Step 3: Express in terms of a single variable and find the minimal $ |m - n| $

Let’s define $ d = m - n $. Then $ m = n + d $. Substitute into the original equation:

$$
3(n + d) + 4n = 100 \quad \Rightarrow \quad 7n + 3d = 100 \quad \Rightarrow \quad n = \frac{100 - 3d}{7}
$$

For $ n $ to be an integer, $ 100 - 3d $ must be divisible by 7. So:

$$
100 - 3d \equiv 0 \pmod{7} \quad \Rightarrow \quad 3d \equiv 100 \pmod{7}
$$

Now, since $ 100 \equiv 2 \pmod{7} $, this gives:

$$
3d \equiv 2 \pmod{7}
$$

The modular inverse of 3 modulo 7 is 5 (since $ 3 \cdot 5 = 15 \equiv 1 \pmod{7} $), so:

$$
d \equiv 2 \cdot 5 = 10 \equiv 3 \pmod{7}
$$

So $ d = 3 + 7k $ for some integer $ k $, and hence:

$$
|m - n| = |d| = |3 + 7k|
$$

We want the **minimum** value of this expression. The closest value to 0 is when $ k = 0 $, giving:

$$
|m - n| = |3| = 3
$$

So, the **smallest possible** value of $ |m - n| $ is:

$$
\boxed{3}
$$

---

### Verification with actual integers

To ensure this is correct, let’s find such $ m $ and $ n $:

If $ d = 3 \Rightarrow m = n + 3 $

Substitute into the equation:

$$
3(n + 3) + 4n = 100 \Rightarrow 3n + 9 + 4n = 100 \Rightarrow 7n = 91 \Rightarrow n = 13
$$

Then $ m = 13 + 3 = 16 $

So $ m = 16, n = 13 \Rightarrow |m - n| = 3 $

No values of $ d $ give a smaller absolute value. Attempts with $ d = 1, 2 $ fail to produce integer solutions. Thus, the minimal is confirmed.

---

### Final Answer:

$$
\boxed{3}
$$

[Assistant]
Explanation:
Okay, so I need to find the smallest possible value of |m - n| where m and n are integers that satisfy the equation 3m + 4n = 100. Hmm, let me think about how to approach this.

First, maybe I can express one variable in terms of the other. Let me try solving for m in terms of n. If I rearrange the equation:

3m = 100 - 4n
So, m = (100 - 4n)/3

Since m has to be an integer, (100 - 4n) must be divisible by 3. That means 100 - 4n ≡ 0 mod 3. Let me write that down:

100 - 4n ≡ 0 mod 3
Which simplifies to -4n ≡ -100 mod 3
Or equivalently, 4n ≡ 100 mod 3

But 4 is congruent to 1 mod 3, so 4n ≡ n mod 3. Similarly, 100 divided by 3 is 33 with a remainder of 1, so 100 ≡ 1 mod 3. Therefore:

n ≡ 1 mod 3
Which means n can be written as n = 3k + 1 for some integer k.

Okay, so n = 3k + 1. Then I can substitute back into the equation for m:

m = (100 - 4*(3k + 1))/3
Let me simplify that:

m = (100 - 12k - 4)/3
m = (96 - 12k)/3
m = 32 - 4k

So, m = 32 - 4k and n = 3k + 1. Now, since m and n are integers, k can be any integer. Now, I need to find the value of k such that |m - n| is minimized.

Let me compute m - n:

m - n = (32 - 4k) - (3k + 1) = 32 - 4k - 3k - 1 = 31 - 7k

So, |m - n| = |31 - 7k|

Therefore, the problem reduces to finding integer k such that |31 - 7k| is as small as possible. Then, the minimal value of |31 - 7k| is the minimal |m - n|.

So, I need to find integer k such that 7k is as close as possible to 31. Let me solve 7k ≈ 31.

Dividing 31 by 7: 7*4 = 28, 7*5 = 35. So 31 is between 28 and 35. Therefore, the closest integers are 28 (k=4) and 35 (k=5). Let me compute |31 - 28| = 3 and |31 - 35| = 4. So the closest is 3 when k=4.

Wait, but maybe there are other k values that could give a smaller difference? Let me check. For example, if k=3, then 7*3=21, |31 -21|=10. If k=4, |31 -28|=3. If k=5, |31 -35|=4. If k=6, |31 -42|=11, which is bigger. Similarly, k=2: 7*2=14, |31-14|=17. So, yes, k=4 gives the minimal value of 3.

But wait, is that the minimal possible? Let me check. Is there a k where 7k is closer to 31? For example, if k=4. Let me see. Since 7k can only take integer multiples, and 31 divided by 7 is approximately 4.428... So the closest integers are 4 and 5. So 4.428 is between 4 and 5, so the closest integer is 4. So, 7*4=28, difference 3, which is smaller than 7*5=35, difference 4. So the minimal |31 -7k| is 3, which would be the minimal |m -n|? So the answer would be 3?

But let me check with actual values of m and n. Let me take k=4.

k=4: n=3*4 +1=13, m=32 -4*4=32 -16=16. So m=16, n=13. Then |16 -13|=3. Okay.

Is there a case where |m -n| is less than 3? Let me check. Let me see if there's another k where |31 -7k| is less than 3. For that, |31 -7k| <3 implies that 31 -7k is between -3 and 3. So:

-3 < 31 -7k <3
Subtract 31: -34 < -7k < -28
Multiply by -1 and reverse inequalities: 28 < 7k <34
Divide by 7: 4 < k < 34/7 ≈4.857

So k must be between 4 and 4.857. Since k is integer, k=4 is the only integer in that interval. Therefore, the minimal value is indeed 3.

Wait, so is there another solution? Wait, but maybe I made an assumption when solving for n? Let me check if there are other solutions where maybe m and n aren't expressed in terms of k like that?

Alternatively, maybe I should approach the problem by checking possible values of n or m?

Alternatively, perhaps I can think about this as a linear Diophantine equation. The equation 3m + 4n = 100. Since 3 and 4 are coprime, there are integer solutions. The general solution can be found once a particular solution is found.

But since I already found expressions for m and n in terms of k, maybe that's sufficient. But just to make sure, let me check if there are other solutions.

Alternatively, suppose we consider the problem of minimizing |m - n|. Let me think about that. Let me denote d = m - n. Then m = n + d. Substitute into the equation:

3(n + d) + 4n = 100
3n + 3d + 4n = 100
7n + 3d = 100
So, 7n = 100 - 3d
Therefore, n = (100 - 3d)/7

Since n must be integer, 100 - 3d must be divisible by 7. Therefore, 100 - 3d ≡ 0 mod 7
Which implies 3d ≡ 100 mod 7
Calculate 100 mod 7: 7*14=98, so 100 ≡ 2 mod 7. Therefore, 3d ≡ 2 mod 7
Multiply both sides by the inverse of 3 mod 7. Since 3*5=15≡1 mod7, so inverse of 3 is 5. Therefore, d ≡ 2*5 ≡10 ≡3 mod7. Therefore, d ≡3 mod7.

Therefore, the possible values of d are integers congruent to 3 mod7. Therefore, d can be ... -4, 3, 10, 17,... or 3, -4, -11, etc. Therefore, the minimal |d| is the minimal among these values. The closest to zero would be d=3 or d=-4. So |3| = 3, |-4|=4. Therefore, the minimal |d| is 3. Therefore, the minimal |m -n| is 3.

Hence, confirming the previous result. Therefore, the answer is 3.

But let me check with actual numbers. If d=3, then n=(100 - 3*3)/7=(100 -9)/7=91/7=13. Therefore, n=13, then m =n +d=13+3=16. Which is the same as before, m=16, n=13. Which gives |16-13|=3. If d=-4, then n=(100 -3*(-4))/7=(100 +12)/7=112/7=16. Then m =16 + (-4)=12. Therefore, m=12, n=16. Then |12-16|=4. Which is larger. Therefore, indeed, the minimal is 3.

Therefore, the answer is 3. Wait, but let me check if there are other values of d with smaller |d|. For example, if d=3-7= -4, which we already checked. What about d=3+7=10? Then |10|=10, which is bigger. Similarly, d=-4-7=-11, |d|=11. So no, the minimal is 3. Therefore, the minimal |m -n| is 3.

Alternatively, let me think if there are other solutions where |m -n| is smaller. Let me check some nearby values.

Suppose |m -n|=1. Let's see if there are integers m and n such that 3m +4n=100 and |m -n|=1. So, either m =n +1 or n =m +1.

Case 1: m =n +1. Then substitute into equation:

3(n +1) +4n =100
3n +3 +4n =100
7n +3 =100
7n=97
n=97/7 ≈13.857... Not integer. So no solution.

Case 2: n =m +1. Substitute into equation:

3m +4(m +1)=100
3m +4m +4=100
7m +4=100
7m=96
m=96/7 ≈13.714... Not integer. So no solution. Therefore, there's no solution with |m -n|=1.

Similarly, check |m -n|=2.

Case 1: m =n +2. Substitute into equation:

3(n +2) +4n = 100
3n +6 +4n =100
7n +6=100
7n=94
n=94/7 ≈13.428... Not integer.

Case 2: n =m +2. Substitute into equation:

3m +4(m +2)=100
3m +4m +8=100
7m +8=100
7m=92
m=92/7 ≈13.142... Not integer.

Therefore, no solution with |m -n|=2. So the minimal possible is indeed 3. Therefore, the answer is 3.

But just to be thorough, let me check if there are solutions for other k values that might result in |m -n| smaller than 3. Wait, earlier when I considered k=4, we had |m -n|=3. Let me check if there are other k values where |31 -7k| is less than 3. For example, if k=4 gives 3, k=3 gives |31 -21|=10, k=5 gives |31 -35|=4, k=2 gives |31 -14|=17, etc. So no, nothing smaller than 3. Therefore, 3 is indeed the minimal.

Alternatively, maybe there's another way to approach this problem. For example, considering the equation 3m +4n=100 and trying to find m and n such that m -n is minimized. Let me think.

Let me consider m -n = d. So m = n + d. Then substituting into original equation:

3(n + d) +4n =100
3n + 3d +4n =100
7n +3d =100
So 7n =100 -3d
Thus, n=(100 -3d)/7. For n to be integer, 100 -3d must be divisible by 7. Which is what I did earlier. So the same conclusion follows.

Alternatively, maybe trying to find solutions by checking possible values? For example, if I fix n and compute m? But since there are infinitely many n, maybe not. But perhaps we can find bounds for n?

Since 3m +4n=100, m and n are integers. Let me find the possible range for n. Let me suppose that m and n are positive integers. Wait, but the problem says integers, not necessarily positive. So m and n can be positive or negative. However, if we consider that, maybe there are solutions with negative numbers that give a smaller |m -n|. Wait, but let's see.

But if I assume that m and n are positive integers, then we can find some bounds. Let me check.

Suppose m and n are positive integers. Then 3m +4n=100. So 3m < 100, so m < 100/3 ≈33.33, so m ≤33. Similarly, 4n < 100, so n <25. So n ≤24. So possible values for n from 1 to 24, and m accordingly. But since we need to minimize |m -n|, maybe it's better to check around the values where m and n are close.

But even if we consider negative integers, maybe there are solutions with |m -n| smaller? Let me check.

But earlier approaches showed that the minimal |d| is 3 regardless of sign. But let me check.

For example, suppose m and n are such that m -n = -3. Then |m -n|=3, same as before. So even if negative, it's same. So maybe the minimal is 3.

But let me try to think if there are solutions where |m -n| is smaller. For example, suppose m -n=0. Then m=n. Then 3m +4m=7m=100. So m=100/7≈14.28... Not integer. So no solution.

If m -n=±1, we saw no solution. Similarly for ±2. So no solution. So 3 is the minimal.

Alternatively, maybe considering the problem in terms of linear combinations and the concept of the greatest common divisor?

The equation is 3m +4n=100. Since gcd(3,4)=1, which divides 100, there are solutions.

But how does that help with minimizing |m -n|? Maybe not directly, but confirms that solutions exist.

Alternatively, think about the problem geometrically? The equation 3m +4n=100 is a straight line in the m-n plane. We need the point (m,n) on this line with integer coordinates that is closest to the line m =n, since |m -n| is the distance from the line m=n. But the minimal distance might not necessarily be an integer point, but since we need integer points, we need to find the closest one. However, this approach might be more complicated. But maybe we can use some formulas?

Alternatively, think about the problem as an optimization problem with constraints.

But perhaps that's overcomplicating. Since my previous approaches have already given me 3 as the minimal value, and verification shows that there are solutions with |m -n|=3, and no solutions with smaller, I think the answer is 3.

But just to be absolutely thorough, let me check a few more values around k=4. For example, k=4 gives n=13, m=16. Let me check k=3: n=3*3+1=10, m=32 -4*3=32 -12=20. Then |20 -10|=10. k=5: n=3*5 +1=16, m=32 -4*5=32 -20=12. |12 -16|=4. k=6: n=19, m=32-24=8. |8 -19|=11. k=2: n=7, m=32 -8=24. |24-7|=17. k=1: n=4, m=32 -4=28. |28 -4|=24. k=0: n=1, m=32. |32 -1|=31. k=-1: n=-2, m=32 - (-4)=36. |36 - (-2)|=38. So as k moves away from 4, the |m -n| increases. So indeed, the minimum is at k=4 with value 3.

Therefore, after all these checks, I can be confident that the minimal |m -n| is 3.

**Final Answer**
\boxed{3}

To determine the smallest possible value of $ |m - n| $, where $ m $ and $ n $ are integers satisfying the equation:

$$
3m + 4n = 100
$$

we proceed as follows:

---

### Step 1: Express $ m $ in terms of $ n $

We solve for $ m $:

$$
3m = 100 - 4n \quad \Rightarrow \quad m = \frac{100 - 4n}{3}
$$

For $ m $ to be an integer, the numerator $ 100 - 4n $ must be divisible by 3. So we examine the modular condition:

$$
100 - 4n \equiv 0 \pmod{3} \quad \Rightarrow \quad -4n \equiv -100 \pmod{3}
$$

Since $-4 \equiv 2 \pmod{3}$ and $-100 \equiv 1 \pmod{3}$, we get:

$$
2n \equiv 1 \pmod{3}
$$

Now, the multiplicative inverse of 2 modulo 3 is 2 (since $ 2 \times 2 = 4 \equiv 1 \pmod{3} $), so:

$$
n \equiv 2 \cdot 1 \equiv 2 \pmod{3}
$$

Thus, we write:

$$
n = 3k + 2 \quad \text{for some integer } k
$$

---

### Step 2: Express $ m $ in terms of $ k $

Substitute $ n = 3k + 2 $ into the expression for $ m $:

$$
m = \frac{100 - 4(3k + 2)}{3} = \frac{100 - 12k - 8}{3} = \frac{92 - 12k}{3} = 30.66\ldots
$$

Wait—this is only valid if the numerator is divisible by 3. Let’s recheck.

Instead, compute $ m $ properly:

$$
m = \frac{100 - 4\left(3k + 2\right)}{3} = \frac{100 - 12k - 8}{3} = \frac{92 - 12k}{3}
$$

Simplify:

$$
m = \frac{92 - 12k}{3} = 30.666\ldots \quad \text{Not an integer unless 92 - 12k is divisible by 3}
$$

Try a different approach. Solve the original equation with **integer constraints** directly.

---

### Step 3: Express in terms of a single variable and find the minimal $ |m - n| $

Let’s define $ d = m - n $. Then $ m = n + d $. Substitute into the original equation:

$$
3(n + d) + 4n = 100 \quad \Rightarrow \quad 7n + 3d = 100 \quad \Rightarrow \quad n = \frac{100 - 3d}{7}
$$

For $ n $ to be an integer, $ 100 - 3d $ must be divisible by 7. So:

$$
100 - 3d \equiv 0 \pmod{7} \quad \Rightarrow \quad 3d \equiv 100 \pmod{7}
$$

Now, since $ 100 \equiv 2 \pmod{7} $, this gives:

$$
3d \equiv 2 \pmod{7}
$$

The modular inverse of 3 modulo 7 is 5 (since $ 3 \cdot 5 = 15 \equiv 1 \pmod{7} $), so:

$$
d \equiv 2 \cdot 5 = 10 \equiv 3 \pmod{7}
$$

So $ d = 3 + 7k $ for some integer $ k $, and hence:

$$
|m - n| = |d| = |3 + 7k|
$$

We want the **minimum** value of this expression. The closest value to 0 is when $ k = 0 $, giving:

$$
|m - n| = |3| = 3
$$

So, the **smallest possible** value of $ |m - n| $ is:

$$
\boxed{3}
$$

---

### Verification with actual integers

To ensure this is correct, let’s find such $ m $ and $ n $:

If $ d = 3 \Rightarrow m = n + 3 $

Substitute into the equation:

$$
3(n + 3) + 4n = 100 \Rightarrow 3n + 9 + 4n = 100 \Rightarrow 7n = 91 \Rightarrow n = 13
$$

Then $ m = 13 + 3 = 16 $

So $ m = 16, n = 13 \Rightarrow |m - n| = 3 $

No values of $ d $ give a smaller absolute value. Attempts with $ d = 1, 2 $ fail to produce integer solutions. Thus, the minimal is confirmed.

---

### Final Answer:

$$
\boxed{3}
$$

Solution:
From $3m + 4n = 100$ we can get that $n = 25 - \frac{3}{4}m$.  Then we want to minimize $$\left| m - n \right| = \left| m - 25 + \frac{3}{4}m \right| =\left|  \frac{7}{4}m - 25 \right| = \left|  7m - 100 \right|$$In other words we want $7m$ as close to $100$ as possible while still giving us integer solutions for $m$ and $n$ to the equation $3m + 4n = 100$.

By trial and error, we can find that the solution to $3m + 4n = 100$ that makes $m$ closest to $\frac{100}{7}$ is $(m,n) = (16,13)$.  Then we have $\left| m - n \right| = 16-13 =\boxed{3}$.


Explanation:
Okay, let me try to figure out the positive square root of the product 10 × 15 × 24. Hmm, so first, I need to calculate the product of these three numbers and then find the square root of that product. But maybe there's a smarter way than just multiplying them all out directly? Let me think.

Let me start by recalling that the square root of a product can sometimes be broken down into the product of square roots. So, √(a × b × c) = √a × √b × √c. But I'm not sure if that helps here. Maybe I can factor each number into primes and see if there are square factors?

Let me try that approach. Let me factor each number:

10 is 2 × 5.

15 is 3 × 5.

24 is 2³ × 3.

So, multiplying them all together: 10 × 15 × 24 = (2 × 5) × (3 × 5) × (2³ × 3). Let me multiply those together.

First, combine all the prime factors:

For 2s: 2^1 (from 10) × 2^3 (from 24) = 2^(1+3) = 2^4.

For 3s: 3^1 (from 15) × 3^1 (from 24) = 3^(1+1) = 3^2.

For 5s: 5^1 (from 10) × 5^1 (from 15) = 5^(1+1) = 5^2.

So altogether, the product is 2^4 × 3^2 × 5^2.

Now, the square root of this product would be √(2^4 × 3^2 × 5^2).

Since square root of a product is the product of square roots, that would be √(2^4) × √(3^2) × √(5^2).

Calculating each term:

√(2^4) is 2^(4/2) = 2² = 4.

√(3²) is 3^(2/2) = 3^1 = 3.

√(5²) is 5^(2/2) = 5^1 = 5.

So multiplying those together: 4 × 3 × 5. Let me compute that.

First, 4 × 3 = 12, then 12 × 5 = 60.

Wait, is that right? Let me check. If I take 10 × 15 × 24 first and then take the square root, do I get 60? Let me verify by calculating the product first.

Calculating the product step by step:

First, 10 × 15. Well, 10 × 15 is 150. Then, 150 × 24. Let me compute 150 × 24.

Breaking it down: 150 × 20 = 3000, and 150 × 4 = 600. So, 3000 + 600 = 3600.

So the product is 3600. Then the square root of 3600 is... Well, I know that 60² = 3600, so √3600 = 60.

Yes! So both methods give me 60. Therefore, the positive square root is 60.

But just to make sure I didn't make a mistake in my prime factorization. Let me check again:

Original numbers: 10, 15, 24.

10 factors into 2 × 5.

15 factors into 3 × 5.

24 factors into 2³ × 3.

So when multiplied together, 2 × 5 × 3 × 5 × 2³ × 3. Combining like terms:

2s: 1 + 3 = 4.

3s: 1 + 1 = 2.

5s: 1 + 1 = 2.

So indeed, 2^4 × 3² × 5². Taking the square root gives 2² × 3 × 5 = 4 × 3 × 5 = 60.

Therefore, the answer is 60.

Alternatively, maybe there is another way to approach this? Let me see.

Alternatively, maybe I can pair numbers to make squares. For example, 10 × 15 × 24. Let me see if any of these numbers can be paired to form squares or something.

Alternatively, maybe group them as (10 × 24) × 15. Let's compute 10 × 24 first. 10 × 24 is 240, then 240 × 15. Hmm, 240 × 15. Let me calculate that. 240 × 10 = 2400, 240 × 5 = 1200, so 2400 + 1200 = 3600. Again, same product. So square root of 3600 is 60.

Alternatively, maybe there's a way to factor the original numbers to make square roots easier. For example, 10 × 15 × 24. Let me see:

Note that 15 is 3 × 5, 10 is 2 × 5, and 24 is 4 × 6. Wait, 24 is 4 × 6. So maybe that's helpful? Then the product is 10 × 15 × 24 = (2 × 5) × (3 × 5) × (4 × 6).

Then, 4 is a square, so maybe taking square roots. Let me think. If I write the product as 2 × 5 × 3 × 5 × 4 × 6. Then grouping 4 × 6? Wait, 4 × 6 is 24. Not sure. Alternatively, let me think of the entire product:

(2 × 5) × (3 × 5) × (4 × 6) = 2 × 5 × 3 × 5 × 4 × 6. Let me rearrange:

Group 4 × 6 = 24. Wait, that's back to where I was before.

Alternatively, note that 4 is 2². So, 2² × 6 × 2 × 5 × 3 × 5. Wait, breaking down:

Original product is 2 × 5 × 3 × 5 × 2² × 6. Then, 2 × 2² is 2³. Wait, maybe that's complicating things.

Alternatively, maybe I can take square root step by step? For example, √(10 × 15 × 24) = √(10 × 15) × √24. Wait, no, that's not correct. Wait, √(a × b × c) = √a × √b × √c. But if I group them as √(10 × 15 × 24) = √( (10 × 15) × 24 ). So √( (10 × 15) × 24 ) = √(10 × 15) × √24.

But if I compute √(10 × 15) first, that would be √150. Then multiplying by √24, which would be √150 × √24. That's √(150 × 24) which is again √3600 = 60. So same result.

Alternatively, maybe simplify √(150 × 24). Let me see if I can factor that. 150 is 25 × 6, and 24 is 4 × 6. Therefore, 150 × 24 = 25 × 6 × 4 × 6 = 25 × 4 × 6 × 6. Then, √(25 × 4 × 6 × 6) = √25 × √4 × √6 × √6 = 5 × 2 × 6 = 5 × 2 × 6 = 60.

Yes! That's another way to break it down. So that's also 60.

So regardless of the method, I end up with 60. So that gives me more confidence that the answer is indeed 60.

Let me just check with another approach. Let me try to compute the product in another way. Maybe 10 × 15 × 24. Let me think of 15 as 10 + 5. Wait, not sure if that helps. Alternatively, 24 is 12 × 2. Let me try:

10 × 15 × 24 = 10 × 15 × 12 × 2. Maybe group 10 × 2 = 20, and 15 × 12 = 180. Then 20 × 180 = 3600. Again, same result. Square root is 60.

Alternatively, think of 10 × 15 × 24. Maybe write them as multiples of 5? Let me see:

10 is 2 × 5, 15 is 3 × 5, and 24 is 24. So the product is 2×5 × 3×5 × 24 = 2×3×5×5×24. Which is 6 × 25 × 24. Then 25 × 24 is 600, so 6 × 600 = 3600. Square root is 60.

Alternatively, I can think of 24 as 5 × 4.8, but that's decimal and might complicate.

Alternatively, maybe think of the square root as sqrt(10*15*24). Let me see if there's a way to simplify the expression under the root. Let me check if there are any common factors. For example:

Compute 10 × 15 = 150, and then 150 × 24. Well, as above. Alternatively, perhaps factor out common terms?

But I think all these methods lead to the same conclusion. So I think 60 is the correct answer.

Just to make sure, let me compute sqrt(10*15*24) numerically. Let me compute 10*15 = 150. 150*24: 150*24. Let me compute 150*24.

Well, 15*24 is 360, so 150*24 is 3600. So sqrt(3600) is 60.

Yes! Therefore, the answer is 60.

I think I've checked this multiple ways, so I feel confident that the positive square root is 60.

**Final Answer**
The positive square root of the product $10 \times 15 \times 24$ is \boxed{60}.

To find the **positive square root** of the product $10 \times 15 \times 24$, we start by analyzing the expression step by step.

---

### Step 1: Factor Each Number into Primes

Let’s factor each of the individual numbers into their prime components:

- $10 = 2 \times 5$
- $15 = 3 \times 5$
- $24 = 2^3 \times 3$

Multiplying these together:

$$
10 \times 15 \times 24 = (2 \times 5) \times (3 \times 5) \times (2^3 \times 3)
$$

Now combine all the prime factors:

- $2^{1 + 3} = 2^4$
- $3^{1 + 1} = 3^2$
- $5^{1 + 1} = 5^2$

So the entire product becomes:

$$
2^4 \times 3^2 \times 5^2
$$

---

### Step 2: Take the Square Root

Now, apply the square root to the entire expression:

$$
\sqrt{2^4 \times 3^2 \times 5^2}
$$

Using the property that $\sqrt{a \cdot b \cdot c} = \sqrt{a} \cdot \sqrt{b} \cdot \sqrt{c}$, we compute each square root:

- $\sqrt{2^4} = 2^2 = 4$
- $\sqrt{3^2} = 3$
- $\sqrt{5^2} = 5$

Multiply them together:

$$
4 \times 3 \times 5 = (4 \times 3) \times 5 = 12 \times 5 = 60
$$

---

### Step 3: Verification (Optional, for Double-Checking)

To ensure accuracy, compute the actual product first:

- $10 \times 15 = 150$
- $150 \times 24 = 3600$

Then take the square root:

$$
\sqrt{3600} = 60
$$

This confirms our previous result.

---

### Final Answer

$$
\boxed{60}
$$

[Assistant]
Explanation:
Okay, let me try to figure out the positive square root of the product 10 × 15 × 24. Hmm, so first, I need to calculate the product of these three numbers and then find the square root of that product. But maybe there's a smarter way than just multiplying them all out directly? Let me think.

Let me start by recalling that the square root of a product can sometimes be broken down into the product of square roots. So, √(a × b × c) = √a × √b × √c. But I'm not sure if that helps here. Maybe I can factor each number into primes and see if there are square factors?

Let me try that approach. Let me factor each number:

10 is 2 × 5.

15 is 3 × 5.

24 is 2³ × 3.

So, multiplying them all together: 10 × 15 × 24 = (2 × 5) × (3 × 5) × (2³ × 3). Let me multiply those together.

First, combine all the prime factors:

For 2s: 2^1 (from 10) × 2^3 (from 24) = 2^(1+3) = 2^4.

For 3s: 3^1 (from 15) × 3^1 (from 24) = 3^(1+1) = 3^2.

For 5s: 5^1 (from 10) × 5^1 (from 15) = 5^(1+1) = 5^2.

So altogether, the product is 2^4 × 3^2 × 5^2.

Now, the square root of this product would be √(2^4 × 3^2 × 5^2).

Since square root of a product is the product of square roots, that would be √(2^4) × √(3^2) × √(5^2).

Calculating each term:

√(2^4) is 2^(4/2) = 2² = 4.

√(3²) is 3^(2/2) = 3^1 = 3.

√(5²) is 5^(2/2) = 5^1 = 5.

So multiplying those together: 4 × 3 × 5. Let me compute that.

First, 4 × 3 = 12, then 12 × 5 = 60.

Wait, is that right? Let me check. If I take 10 × 15 × 24 first and then take the square root, do I get 60? Let me verify by calculating the product first.

Calculating the product step by step:

First, 10 × 15. Well, 10 × 15 is 150. Then, 150 × 24. Let me compute 150 × 24.

Breaking it down: 150 × 20 = 3000, and 150 × 4 = 600. So, 3000 + 600 = 3600.

So the product is 3600. Then the square root of 3600 is... Well, I know that 60² = 3600, so √3600 = 60.

Yes! So both methods give me 60. Therefore, the positive square root is 60.

But just to make sure I didn't make a mistake in my prime factorization. Let me check again:

Original numbers: 10, 15, 24.

10 factors into 2 × 5.

15 factors into 3 × 5.

24 factors into 2³ × 3.

So when multiplied together, 2 × 5 × 3 × 5 × 2³ × 3. Combining like terms:

2s: 1 + 3 = 4.

3s: 1 + 1 = 2.

5s: 1 + 1 = 2.

So indeed, 2^4 × 3² × 5². Taking the square root gives 2² × 3 × 5 = 4 × 3 × 5 = 60.

Therefore, the answer is 60.

Alternatively, maybe there is another way to approach this? Let me see.

Alternatively, maybe I can pair numbers to make squares. For example, 10 × 15 × 24. Let me see if any of these numbers can be paired to form squares or something.

Alternatively, maybe group them as (10 × 24) × 15. Let's compute 10 × 24 first. 10 × 24 is 240, then 240 × 15. Hmm, 240 × 15. Let me calculate that. 240 × 10 = 2400, 240 × 5 = 1200, so 2400 + 1200 = 3600. Again, same product. So square root of 3600 is 60.

Alternatively, maybe there's a way to factor the original numbers to make square roots easier. For example, 10 × 15 × 24. Let me see:

Note that 15 is 3 × 5, 10 is 2 × 5, and 24 is 4 × 6. Wait, 24 is 4 × 6. So maybe that's helpful? Then the product is 10 × 15 × 24 = (2 × 5) × (3 × 5) × (4 × 6).

Then, 4 is a square, so maybe taking square roots. Let me think. If I write the product as 2 × 5 × 3 × 5 × 4 × 6. Then grouping 4 × 6? Wait, 4 × 6 is 24. Not sure. Alternatively, let me think of the entire product:

(2 × 5) × (3 × 5) × (4 × 6) = 2 × 5 × 3 × 5 × 4 × 6. Let me rearrange:

Group 4 × 6 = 24. Wait, that's back to where I was before.

Alternatively, note that 4 is 2². So, 2² × 6 × 2 × 5 × 3 × 5. Wait, breaking down:

Original product is 2 × 5 × 3 × 5 × 2² × 6. Then, 2 × 2² is 2³. Wait, maybe that's complicating things.

Alternatively, maybe I can take square root step by step? For example, √(10 × 15 × 24) = √(10 × 15) × √24. Wait, no, that's not correct. Wait, √(a × b × c) = √a × √b × √c. But if I group them as √(10 × 15 × 24) = √( (10 × 15) × 24 ). So √( (10 × 15) × 24 ) = √(10 × 15) × √24.

But if I compute √(10 × 15) first, that would be √150. Then multiplying by √24, which would be √150 × √24. That's √(150 × 24) which is again √3600 = 60. So same result.

Alternatively, maybe simplify √(150 × 24). Let me see if I can factor that. 150 is 25 × 6, and 24 is 4 × 6. Therefore, 150 × 24 = 25 × 6 × 4 × 6 = 25 × 4 × 6 × 6. Then, √(25 × 4 × 6 × 6) = √25 × √4 × √6 × √6 = 5 × 2 × 6 = 5 × 2 × 6 = 60.

Yes! That's another way to break it down. So that's also 60.

So regardless of the method, I end up with 60. So that gives me more confidence that the answer is indeed 60.

Let me just check with another approach. Let me try to compute the product in another way. Maybe 10 × 15 × 24. Let me think of 15 as 10 + 5. Wait, not sure if that helps. Alternatively, 24 is 12 × 2. Let me try:

10 × 15 × 24 = 10 × 15 × 12 × 2. Maybe group 10 × 2 = 20, and 15 × 12 = 180. Then 20 × 180 = 3600. Again, same result. Square root is 60.

Alternatively, think of 10 × 15 × 24. Maybe write them as multiples of 5? Let me see:

10 is 2 × 5, 15 is 3 × 5, and 24 is 24. So the product is 2×5 × 3×5 × 24 = 2×3×5×5×24. Which is 6 × 25 × 24. Then 25 × 24 is 600, so 6 × 600 = 3600. Square root is 60.

Alternatively, I can think of 24 as 5 × 4.8, but that's decimal and might complicate.

Alternatively, maybe think of the square root as sqrt(10*15*24). Let me see if there's a way to simplify the expression under the root. Let me check if there are any common factors. For example:

Compute 10 × 15 = 150, and then 150 × 24. Well, as above. Alternatively, perhaps factor out common terms?

But I think all these methods lead to the same conclusion. So I think 60 is the correct answer.

Just to make sure, let me compute sqrt(10*15*24) numerically. Let me compute 10*15 = 150. 150*24: 150*24. Let me compute 150*24.

Well, 15*24 is 360, so 150*24 is 3600. So sqrt(3600) is 60.

Yes! Therefore, the answer is 60.

I think I've checked this multiple ways, so I feel confident that the positive square root is 60.

**Final Answer**
The positive square root of the product $10 \times 15 \times 24$ is \boxed{60}.

To find the **positive square root** of the product $10 \times 15 \times 24$, we start by analyzing the expression step by step.

---

### Step 1: Factor Each Number into Primes

Let’s factor each of the individual numbers into their prime components:

- $10 = 2 \times 5$
- $15 = 3 \times 5$
- $24 = 2^3 \times 3$

Multiplying these together:

$$
10 \times 15 \times 24 = (2 \times 5) \times (3 \times 5) \times (2^3 \times 3)
$$

Now combine all the prime factors:

- $2^{1 + 3} = 2^4$
- $3^{1 + 1} = 3^2$
- $5^{1 + 1} = 5^2$

So the entire product becomes:

$$
2^4 \times 3^2 \times 5^2
$$

---

### Step 2: Take the Square Root

Now, apply the square root to the entire expression:

$$
\sqrt{2^4 \times 3^2 \times 5^2}
$$

Using the property that $\sqrt{a \cdot b \cdot c} = \sqrt{a} \cdot \sqrt{b} \cdot \sqrt{c}$, we compute each square root:

- $\sqrt{2^4} = 2^2 = 4$
- $\sqrt{3^2} = 3$
- $\sqrt{5^2} = 5$

Multiply them together:

$$
4 \times 3 \times 5 = (4 \times 3) \times 5 = 12 \times 5 = 60
$$

---

### Step 3: Verification (Optional, for Double-Checking)

To ensure accuracy, compute the actual product first:

- $10 \times 15 = 150$
- $150 \times 24 = 3600$

Then take the square root:

$$
\sqrt{3600} = 60
$$

This confirms our previous result.

---

### Final Answer

$$
\boxed{60}
$$

Solution:
We can see that

\begin{align*}
\sqrt{10\cdot 15\cdot 24} &= \sqrt{(2\cdot 5)\cdot (3\cdot 5)\cdot (2^3\cdot 3)}\\
&= \sqrt{2^4\cdot3^2\cdot 5^2} \\
&= 2^2\cdot3\cdot5 \\
&= \boxed{60}.
\end{align*}


Explanation:
Okay, let me try to figure out this problem. So, the question is: Two-thirds of the students at Baker Middle School take music. There are 834 students who take music. How many students are there at Baker Middle School?

Alright, let's break this down. Let me start by understanding what's given and what's being asked.

We know that two-thirds of the total number of students take music, and that number is 834. So, we need to find the total number of students. Let me represent the total number of students as a variable to make it easier. Let's say the total number of students is "x".

So, according to the problem, two-thirds of x is equal to 834. In mathematical terms, that would be (2/3) * x = 834.

Now, I need to solve for x. To do that, I should isolate x on one side of the equation. Since x is being multiplied by 2/3, I can do the opposite operation, which is to divide by 2/3, or multiply by the reciprocal of 2/3. The reciprocal of 2/3 is 3/2.

So, multiplying both sides of the equation by 3/2 should give me x by itself. Let me write that out:

x = 834 * (3/2)

Hmm, let me check if that makes sense. If I take two-thirds of the total students and get 834, then multiplying 834 by 3/2 should give me the total number of students. That seems right because if you think of it as dividing the music students into two parts, each part would be 834 divided by 2, and then multiplying by 3 to get the total.

Let me do the calculation step by step. First, let me compute 834 divided by 2.

834 divided by 2 is 417. Then, multiplying that by 3.

417 * 3. Let me compute that. 400 * 3 is 1200, and 17 * 3 is 51. So adding those together, 1200 + 51 = 1251.

Therefore, the total number of students at Baker Middle School should be 1251.

Wait, let me verify that. Let me check if two-thirds of 1251 is indeed 834.

Calculating two-thirds of 1251.

First, divide 1251 by 3. 1251 divided by 3. Let me do that division. 3 goes into 12 four times, 4*3=12, subtract, bring down 5. 3 goes into 5 once, 1*3=3, subtract, get 2. Bring down 5, making 25. 3 goes into 25 eight times, 8*3=24, subtract, get 1. Bring down 1, making 11. 3 goes into 11 three times, 3*3=9, subtract, get 2. So, 1251 divided by 3 is 417.

Then, multiplying 417 by 2 gives 834.

Yes, that checks out. So, two-thirds of 1251 is 834, which matches the given information. Therefore, the total number of students must be 1251.

Alternatively, another way to think about it is if 834 is two-thirds, then one-third would be half of that, which is 417, and then the total is three times that, which is 1251. Either way, I arrive at the same answer.

I think that's solid. Let me just make sure I didn't make any arithmetic errors.

Starting again:

Total students = x

(2/3)x = 834

Multiply both sides by 3: 2x = 834 * 3

Then divide both sides by 2: x = (834 * 3)/2

Calculating 834 * 3:

Let me compute 800*3 = 2400

34*3 = 102

So total is 2400 + 102 = 2502

Then divide by 2: 2502 / 2 = 1251

Yes, that's the same result.

Alternatively, 834 divided by 2 is 417, then times 3 is 1251. Yep.

So, all methods give me 1251, and checking by plugging back in works.

Therefore, I'm confident that the total number of students at Baker Middle School is 1251.

**Final Answer**
The total number of students at Baker Middle School is \boxed{1251}.

To find the total number of students at Baker Middle School, we are told that two-thirds of the students take music, and there are 834 students who take music.

---

### Step 1: Define the unknown

Let the **total number of students** be represented as $ x $.
We are given that:
$$
\frac{2}{3} \times x = 834
$$

---

### Step 2: Solve the equation

To isolate $ x $, multiply both sides of the equation by the **reciprocal** of $ \frac{2}{3} $, which is $ \frac{3}{2} $:

$$
x = 834 \times \frac{3}{2}
$$

Now perform the multiplication:

First, calculate:
$$
834 \div 2 = 417
$$

Then multiply:
$$
417 \times 3 = 1251
$$

---

### Step 3: Verify the result

To verify, calculate $ \frac{2}{3} \times 1251 $:

First:
$$
1251 \div 3 = 417
$$
Then:
$$
417 \times 2 = 834
$$

This confirms that our original condition is satisfied: 834 students take music, which is two-thirds of the total number of students.

---

### Final Answer

$$
\boxed{1251}
$$

[Assistant]
Explanation:
Okay, let me try to figure out this problem. So, the question is: Two-thirds of the students at Baker Middle School take music. There are 834 students who take music. How many students are there at Baker Middle School?

Alright, let's break this down. Let me start by understanding what's given and what's being asked.

We know that two-thirds of the total number of students take music, and that number is 834. So, we need to find the total number of students. Let me represent the total number of students as a variable to make it easier. Let's say the total number of students is "x".

So, according to the problem, two-thirds of x is equal to 834. In mathematical terms, that would be (2/3) * x = 834.

Now, I need to solve for x. To do that, I should isolate x on one side of the equation. Since x is being multiplied by 2/3, I can do the opposite operation, which is to divide by 2/3, or multiply by the reciprocal of 2/3. The reciprocal of 2/3 is 3/2.

So, multiplying both sides of the equation by 3/2 should give me x by itself. Let me write that out:

x = 834 * (3/2)

Hmm, let me check if that makes sense. If I take two-thirds of the total students and get 834, then multiplying 834 by 3/2 should give me the total number of students. That seems right because if you think of it as dividing the music students into two parts, each part would be 834 divided by 2, and then multiplying by 3 to get the total.

Let me do the calculation step by step. First, let me compute 834 divided by 2.

834 divided by 2 is 417. Then, multiplying that by 3.

417 * 3. Let me compute that. 400 * 3 is 1200, and 17 * 3 is 51. So adding those together, 1200 + 51 = 1251.

Therefore, the total number of students at Baker Middle School should be 1251.

Wait, let me verify that. Let me check if two-thirds of 1251 is indeed 834.

Calculating two-thirds of 1251.

First, divide 1251 by 3. 1251 divided by 3. Let me do that division. 3 goes into 12 four times, 4*3=12, subtract, bring down 5. 3 goes into 5 once, 1*3=3, subtract, get 2. Bring down 5, making 25. 3 goes into 25 eight times, 8*3=24, subtract, get 1. Bring down 1, making 11. 3 goes into 11 three times, 3*3=9, subtract, get 2. So, 1251 divided by 3 is 417.

Then, multiplying 417 by 2 gives 834.

Yes, that checks out. So, two-thirds of 1251 is 834, which matches the given information. Therefore, the total number of students must be 1251.

Alternatively, another way to think about it is if 834 is two-thirds, then one-third would be half of that, which is 417, and then the total is three times that, which is 1251. Either way, I arrive at the same answer.

I think that's solid. Let me just make sure I didn't make any arithmetic errors.

Starting again:

Total students = x

(2/3)x = 834

Multiply both sides by 3: 2x = 834 * 3

Then divide both sides by 2: x = (834 * 3)/2

Calculating 834 * 3:

Let me compute 800*3 = 2400

34*3 = 102

So total is 2400 + 102 = 2502

Then divide by 2: 2502 / 2 = 1251

Yes, that's the same result.

Alternatively, 834 divided by 2 is 417, then times 3 is 1251. Yep.

So, all methods give me 1251, and checking by plugging back in works.

Therefore, I'm confident that the total number of students at Baker Middle School is 1251.

**Final Answer**
The total number of students at Baker Middle School is \boxed{1251}.

To find the total number of students at Baker Middle School, we are told that two-thirds of the students take music, and there are 834 students who take music.

---

### Step 1: Define the unknown

Let the **total number of students** be represented as $ x $.
We are given that:
$$
\frac{2}{3} \times x = 834
$$

---

### Step 2: Solve the equation

To isolate $ x $, multiply both sides of the equation by the **reciprocal** of $ \frac{2}{3} $, which is $ \frac{3}{2} $:

$$
x = 834 \times \frac{3}{2}
$$

Now perform the multiplication:

First, calculate:
$$
834 \div 2 = 417
$$

Then multiply:
$$
417 \times 3 = 1251
$$

---

### Step 3: Verify the result

To verify, calculate $ \frac{2}{3} \times 1251 $:

First:
$$
1251 \div 3 = 417
$$
Then:
$$
417 \times 2 = 834
$$

This confirms that our original condition is satisfied: 834 students take music, which is two-thirds of the total number of students.

---

### Final Answer

$$
\boxed{1251}
$$

Solution: If there are $S$ students at Baker Middle School, then $\frac{2}{3}S$ students take music.  Setting $\frac{2}{3}S$ equal to 834 and multiplying both sides by $\frac{3}{2}$, we find that there are $\frac{3}{2}\times 834=\boxed{1251}$ students at the school.


Explanation:
Okay, so I need to find the value of $ b $ when the point $ (a, b) $ is on the line given by the equation $ 3x + 2y = 12 $, and they told me that $ a = 4 $. Let me think about how to approach this.

First, I remember that if a point lies on a line, then its coordinates must satisfy the equation of that line. That means if I substitute the $ x $ and $ y $ values of the point into the equation, it should hold true. In this case, the point is $ (a, b) $, which translates to $ x = a $ and $ y = b $.

So, since $ a = 4 $, I can substitute that into the equation. Let me write that down:

Original equation: $ 3x + 2y = 12 $

Substituting $ x = 4 $:

$ 3(4) + 2y = 12 $

Let me compute $ 3 \times 4 $. Hmm, 3 times 4 is 12. So that becomes:

$ 12 + 2y = 12 $

Now I need to solve for $ y $, which is $ b $ in this case. Let me subtract 12 from both sides of the equation to isolate the term with $ y $:

$ 12 + 2y - 12 = 12 - 12 $

Simplifying both sides:

On the left side, 12 minus 12 cancels out, leaving $ 2y $. On the right side, 12 minus 12 is also 0. So:

$ 2y = 0 $

To solve for $ y $, I divide both sides by 2:

$ y = 0 / 2 $

Which simplifies to:

$ y = 0 $

Therefore, $ b = 0 $.

Wait, let me check if that makes sense. If $ a = 4 $ and $ b = 0 $, then plugging back into the original equation:

Left side: $ 3(4) + 2(0) = 12 + 0 = 12 $

Right side: 12. Yep, that matches. So it checks out.

But let me think again to make sure I didn't skip any steps or make any calculation errors. The original equation is linear, so there should be exactly one solution for $ y $ when $ x $ is given. Since the equation is $ 3x + 2y = 12 $, solving for $ y $ when $ x = 4 $:

$ 3(4) + 2y = 12 $

$ 12 + 2y = 12 $

Subtract 12: $ 2y = 0 $. Yep, that's correct. Dividing by 2 gives $ y = 0 $.

Another way to think about it is rearranging the original equation to solve for $ y $ in terms of $ x $. Let me try that. Starting with:

$ 3x + 2y = 12 $

Subtract $ 3x $ from both sides:

$ 2y = 12 - 3x $

Divide both sides by 2:

$ y = (12 - 3x)/2 $

So, if $ x = 4 $, then:

$ y = (12 - 3(4))/2 $

Calculating the numerator:

$ 12 - 12 = 0 $, so $ y = 0/2 = 0 $.

Same result. Okay, so that confirms it again.

I guess another check could be to graph the line mentally. The line $ 3x + 2y = 12 $ can be rewritten in slope-intercept form $ y = mx + c $, which might help visualize. Let me do that.

Starting from $ 3x + 2y = 12 $:

Subtract $ 3x $: $ 2y = -3x + 12 $

Divide by 2: $ y = (-3/2)x + 6 $

So, the slope is -3/2 and the y-intercept is 6.

If I imagine this line, when $ x = 0 $, $ y = 6 $, so the point (0, 6) is on the line. When $ y = 0 $, solving for x: $ 3x = 12 $, so $ x = 4 $. Wait, that gives the x-intercept at (4, 0). So, the point (4, 0) is on the line. Which is exactly the point we found when $ a = 4 $, which gives $ b = 0 $. So, that matches.

Therefore, all the methods I used—direct substitution, rearranging the equation, and interpreting the intercepts—lead me to the same conclusion.

I think that's thorough enough. I don't see any mistakes in my reasoning. All checks confirm that $ b = 0 $ when $ a = 4 $.

**Final Answer**
The value of $ b $ is \boxed{0}.

To find the value of $ b $ when the point $ (a, b) $ lies on the line given by the equation:

$$
3x + 2y = 12
$$

and it's given that $ a = 4 $, we can approach this step-by-step:

---

### Step 1: Understand the Relationship
A point $ (x, y) $ lies on the line if substituting $ x $ and $ y $ into the equation satisfies it. In this case, the point is $ (a, b) $, so we substitute $ x = a $ and $ y = b $ into the equation.

---

### Step 2: Substitute $ a = 4 $ into the Equation
Given $ a = 4 $, substitute $ x = 4 $ into the equation:

$$
3(4) + 2b = 12
$$

$$
12 + 2b = 12
$$

---

### Step 3: Solve for $ b $
Subtract 12 from both sides:

$$
2b = 0
$$

Divide both sides by 2:

$$
b = 0
$$

---

### Step 4: Verify the Solution
To confirm, plug $ a = 4 $ and $ b = 0 $ back into the original equation:

$$
3(4) + 2(0) = 12 + 0 = 12
$$

The equation is satisfied, confirming that the point $ (4, 0) $ lies on the line.

---

### Final Result

The value of $ b $ is:

$$
\boxed{0}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of $ b $ when the point $ (a, b) $ is on the line given by the equation $ 3x + 2y = 12 $, and they told me that $ a = 4 $. Let me think about how to approach this.

First, I remember that if a point lies on a line, then its coordinates must satisfy the equation of that line. That means if I substitute the $ x $ and $ y $ values of the point into the equation, it should hold true. In this case, the point is $ (a, b) $, which translates to $ x = a $ and $ y = b $.

So, since $ a = 4 $, I can substitute that into the equation. Let me write that down:

Original equation: $ 3x + 2y = 12 $

Substituting $ x = 4 $:

$ 3(4) + 2y = 12 $

Let me compute $ 3 \times 4 $. Hmm, 3 times 4 is 12. So that becomes:

$ 12 + 2y = 12 $

Now I need to solve for $ y $, which is $ b $ in this case. Let me subtract 12 from both sides of the equation to isolate the term with $ y $:

$ 12 + 2y - 12 = 12 - 12 $

Simplifying both sides:

On the left side, 12 minus 12 cancels out, leaving $ 2y $. On the right side, 12 minus 12 is also 0. So:

$ 2y = 0 $

To solve for $ y $, I divide both sides by 2:

$ y = 0 / 2 $

Which simplifies to:

$ y = 0 $

Therefore, $ b = 0 $.

Wait, let me check if that makes sense. If $ a = 4 $ and $ b = 0 $, then plugging back into the original equation:

Left side: $ 3(4) + 2(0) = 12 + 0 = 12 $

Right side: 12. Yep, that matches. So it checks out.

But let me think again to make sure I didn't skip any steps or make any calculation errors. The original equation is linear, so there should be exactly one solution for $ y $ when $ x $ is given. Since the equation is $ 3x + 2y = 12 $, solving for $ y $ when $ x = 4 $:

$ 3(4) + 2y = 12 $

$ 12 + 2y = 12 $

Subtract 12: $ 2y = 0 $. Yep, that's correct. Dividing by 2 gives $ y = 0 $.

Another way to think about it is rearranging the original equation to solve for $ y $ in terms of $ x $. Let me try that. Starting with:

$ 3x + 2y = 12 $

Subtract $ 3x $ from both sides:

$ 2y = 12 - 3x $

Divide both sides by 2:

$ y = (12 - 3x)/2 $

So, if $ x = 4 $, then:

$ y = (12 - 3(4))/2 $

Calculating the numerator:

$ 12 - 12 = 0 $, so $ y = 0/2 = 0 $.

Same result. Okay, so that confirms it again.

I guess another check could be to graph the line mentally. The line $ 3x + 2y = 12 $ can be rewritten in slope-intercept form $ y = mx + c $, which might help visualize. Let me do that.

Starting from $ 3x + 2y = 12 $:

Subtract $ 3x $: $ 2y = -3x + 12 $

Divide by 2: $ y = (-3/2)x + 6 $

So, the slope is -3/2 and the y-intercept is 6.

If I imagine this line, when $ x = 0 $, $ y = 6 $, so the point (0, 6) is on the line. When $ y = 0 $, solving for x: $ 3x = 12 $, so $ x = 4 $. Wait, that gives the x-intercept at (4, 0). So, the point (4, 0) is on the line. Which is exactly the point we found when $ a = 4 $, which gives $ b = 0 $. So, that matches.

Therefore, all the methods I used—direct substitution, rearranging the equation, and interpreting the intercepts—lead me to the same conclusion.

I think that's thorough enough. I don't see any mistakes in my reasoning. All checks confirm that $ b = 0 $ when $ a = 4 $.

**Final Answer**
The value of $ b $ is \boxed{0}.

To find the value of $ b $ when the point $ (a, b) $ lies on the line given by the equation:

$$
3x + 2y = 12
$$

and it's given that $ a = 4 $, we can approach this step-by-step:

---

### Step 1: Understand the Relationship
A point $ (x, y) $ lies on the line if substituting $ x $ and $ y $ into the equation satisfies it. In this case, the point is $ (a, b) $, so we substitute $ x = a $ and $ y = b $ into the equation.

---

### Step 2: Substitute $ a = 4 $ into the Equation
Given $ a = 4 $, substitute $ x = 4 $ into the equation:

$$
3(4) + 2b = 12
$$

$$
12 + 2b = 12
$$

---

### Step 3: Solve for $ b $
Subtract 12 from both sides:

$$
2b = 0
$$

Divide both sides by 2:

$$
b = 0
$$

---

### Step 4: Verify the Solution
To confirm, plug $ a = 4 $ and $ b = 0 $ back into the original equation:

$$
3(4) + 2(0) = 12 + 0 = 12
$$

The equation is satisfied, confirming that the point $ (4, 0) $ lies on the line.

---

### Final Result

The value of $ b $ is:

$$
\boxed{0}
$$

Solution:
We plug in $x = 4$: \begin{align*}
3(4) + 2y &= 12\\
12 + 2y &= 12\\
y &= 0.
\end{align*}

Therefore, $b = \boxed{0}$.


Explanation:
Okay, so I need to find the value of x such that the arithmetic mean of the numbers 7, 2, x, and 10 is 9. Hmm, arithmetic mean... right, that's the sum of the numbers divided by how many numbers there are. Let me recall the formula for the arithmetic mean.

The arithmetic mean of a set of numbers is calculated by adding all the numbers together and then dividing by the count of numbers. In this case, there are four numbers: 7, 2, x, and 10. The mean is given as 9. So, if I set up the equation, it should be (7 + 2 + x + 10) divided by 4 equals 9. Let me write that down to visualize it better:

(7 + 2 + x + 10) / 4 = 9

Alright, so I need to solve for x. Let me simplify the left side first. Let me add up the numbers that don't involve x. 7 plus 2 is 9, and then 9 plus 10 is 19. Wait, no. Wait, hold on. Let me check that again.

Wait, 7 + 2 is 9, then adding 10 would be 9 + 10 = 19? Wait, but hold on, there's also the x. Let me make sure I do this correctly. The total sum is 7 + 2 + x + 10. So, adding 7 + 2 gives 9, then adding 10 gives 9 + 10 = 19, and then we still have the x. So the total sum is 19 + x. Therefore, the equation becomes:

(19 + x) / 4 = 9

Okay, that seems right. Let me confirm. If I add 7, 2, x, and 10, then that's 7 + 2 + x + 10. Yes, that is (7 + 2 + 10) + x, which is 19 + x. So the equation is correct.

Now, I need to solve for x. To do that, I can multiply both sides of the equation by 4 to eliminate the denominator. Let me do that:

(19 + x) / 4 = 9
Multiply both sides by 4:
19 + x = 9 * 4
Calculating the right side: 9 * 4 is 36.
So, 19 + x = 36

Now, to solve for x, subtract 19 from both sides:

x = 36 - 19

Let me compute 36 - 19. Hmm, 36 minus 20 would be 16, but since it's 19, it's 16 + 1 = 17? Wait, no. Wait, 36 - 19. Let me think again. 19 + 17 is 36? Let me check: 19 + 10 is 29, 29 + 7 is 36. Yes, so 19 + 17 = 36. Therefore, x = 17.

Wait, but let me verify that again. If x is 17, then the numbers are 7, 2, 17, 10. Let me add them up: 7 + 2 = 9, 9 + 17 = 26, 26 + 10 = 36. Then, divide by 4: 36 / 4 = 9. Yes, that's correct. So the arithmetic mean is indeed 9.

Wait, so my calculation seems correct. But let me check if I made any mistake in the steps. Starting from the equation:

(7 + 2 + x + 10)/4 = 9.

Adding 7 + 2 + 10: 7 + 2 is 9, 9 + 10 is 19. So 19 + x over 4 equals 9. Multiply both sides by 4: 19 + x = 36. Subtract 19: x = 17. Yep, that seems right.

Is there another way to approach this problem? Maybe by thinking about what the total sum should be? Since the mean is 9 and there are 4 numbers, the total sum should be 9*4=36. Then, the sum of the given numbers is 7 + 2 + 10 + x. Which is 19 + x. So 19 + x = 36. Therefore, x = 36 - 19 = 17. Yep, same result.

So, I think that's solid. I don't see any errors in my reasoning. Let me check again with the numbers. If x is 17, the numbers are 7, 2, 17, 10. Adding them up: 7 + 2 is 9, 9 + 17 is 26, 26 + 10 is 36. 36 divided by 4 is 9, which matches the given mean. So that checks out.

Therefore, the value of x must be 17. I think that's the correct answer.

**Final Answer**
The value of $ x $ is \boxed{17}.

To determine the value of $ x $, we are given that the arithmetic mean of the numbers 7, 2, $ x $, and 10 is 9. The arithmetic mean of a set of numbers is calculated as the sum of the numbers divided by the total count of numbers.

---

### Step 1: Set Up the Equation

The formula for the arithmetic mean is:

$$
\text{Mean} = \frac{\text{Sum of the numbers}}{\text{Number of terms}}
$$

Given:
- Mean = 9
- Number of terms = 4
- Numbers = 7, 2, $ x $, and 10

So the equation becomes:

$$
\frac{7 + 2 + x + 10}{4} = 9
$$

---

### Step 2: Simplify the Equation

Add the known numbers:

$$
7 + 2 + 10 = 19
$$

So the equation becomes:

$$
\frac{19 + x}{4} = 9
$$

---

### Step 3: Solve for $ x $

Multiply both sides of the equation by 4 to eliminate the denominator:

$$
19 + x = 9 \times 4 = 36
$$

Now solve for $ x $ by subtracting 19 from both sides:

$$
x = 36 - 19 = 17
$$

---

### Step 4: Verify the Solution

To ensure the solution is correct, substitute $ x = 17 $ back into the original set of numbers:

$$
\{7, 2, 17, 10\}
$$

Sum of the numbers:

$$
7 + 2 + 17 + 10 = 36
$$

Divide by the number of terms (4):

$$
\frac{36}{4} = 9
$$

The mean is indeed 9, confirming the solution is correct.

---

### Final Answer

$$
\boxed{17}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of x such that the arithmetic mean of the numbers 7, 2, x, and 10 is 9. Hmm, arithmetic mean... right, that's the sum of the numbers divided by how many numbers there are. Let me recall the formula for the arithmetic mean.

The arithmetic mean of a set of numbers is calculated by adding all the numbers together and then dividing by the count of numbers. In this case, there are four numbers: 7, 2, x, and 10. The mean is given as 9. So, if I set up the equation, it should be (7 + 2 + x + 10) divided by 4 equals 9. Let me write that down to visualize it better:

(7 + 2 + x + 10) / 4 = 9

Alright, so I need to solve for x. Let me simplify the left side first. Let me add up the numbers that don't involve x. 7 plus 2 is 9, and then 9 plus 10 is 19. Wait, no. Wait, hold on. Let me check that again.

Wait, 7 + 2 is 9, then adding 10 would be 9 + 10 = 19? Wait, but hold on, there's also the x. Let me make sure I do this correctly. The total sum is 7 + 2 + x + 10. So, adding 7 + 2 gives 9, then adding 10 gives 9 + 10 = 19, and then we still have the x. So the total sum is 19 + x. Therefore, the equation becomes:

(19 + x) / 4 = 9

Okay, that seems right. Let me confirm. If I add 7, 2, x, and 10, then that's 7 + 2 + x + 10. Yes, that is (7 + 2 + 10) + x, which is 19 + x. So the equation is correct.

Now, I need to solve for x. To do that, I can multiply both sides of the equation by 4 to eliminate the denominator. Let me do that:

(19 + x) / 4 = 9
Multiply both sides by 4:
19 + x = 9 * 4
Calculating the right side: 9 * 4 is 36.
So, 19 + x = 36

Now, to solve for x, subtract 19 from both sides:

x = 36 - 19

Let me compute 36 - 19. Hmm, 36 minus 20 would be 16, but since it's 19, it's 16 + 1 = 17? Wait, no. Wait, 36 - 19. Let me think again. 19 + 17 is 36? Let me check: 19 + 10 is 29, 29 + 7 is 36. Yes, so 19 + 17 = 36. Therefore, x = 17.

Wait, but let me verify that again. If x is 17, then the numbers are 7, 2, 17, 10. Let me add them up: 7 + 2 = 9, 9 + 17 = 26, 26 + 10 = 36. Then, divide by 4: 36 / 4 = 9. Yes, that's correct. So the arithmetic mean is indeed 9.

Wait, so my calculation seems correct. But let me check if I made any mistake in the steps. Starting from the equation:

(7 + 2 + x + 10)/4 = 9.

Adding 7 + 2 + 10: 7 + 2 is 9, 9 + 10 is 19. So 19 + x over 4 equals 9. Multiply both sides by 4: 19 + x = 36. Subtract 19: x = 17. Yep, that seems right.

Is there another way to approach this problem? Maybe by thinking about what the total sum should be? Since the mean is 9 and there are 4 numbers, the total sum should be 9*4=36. Then, the sum of the given numbers is 7 + 2 + 10 + x. Which is 19 + x. So 19 + x = 36. Therefore, x = 36 - 19 = 17. Yep, same result.

So, I think that's solid. I don't see any errors in my reasoning. Let me check again with the numbers. If x is 17, the numbers are 7, 2, 17, 10. Adding them up: 7 + 2 is 9, 9 + 17 is 26, 26 + 10 is 36. 36 divided by 4 is 9, which matches the given mean. So that checks out.

Therefore, the value of x must be 17. I think that's the correct answer.

**Final Answer**
The value of $ x $ is \boxed{17}.

To determine the value of $ x $, we are given that the arithmetic mean of the numbers 7, 2, $ x $, and 10 is 9. The arithmetic mean of a set of numbers is calculated as the sum of the numbers divided by the total count of numbers.

---

### Step 1: Set Up the Equation

The formula for the arithmetic mean is:

$$
\text{Mean} = \frac{\text{Sum of the numbers}}{\text{Number of terms}}
$$

Given:
- Mean = 9
- Number of terms = 4
- Numbers = 7, 2, $ x $, and 10

So the equation becomes:

$$
\frac{7 + 2 + x + 10}{4} = 9
$$

---

### Step 2: Simplify the Equation

Add the known numbers:

$$
7 + 2 + 10 = 19
$$

So the equation becomes:

$$
\frac{19 + x}{4} = 9
$$

---

### Step 3: Solve for $ x $

Multiply both sides of the equation by 4 to eliminate the denominator:

$$
19 + x = 9 \times 4 = 36
$$

Now solve for $ x $ by subtracting 19 from both sides:

$$
x = 36 - 19 = 17
$$

---

### Step 4: Verify the Solution

To ensure the solution is correct, substitute $ x = 17 $ back into the original set of numbers:

$$
\{7, 2, 17, 10\}
$$

Sum of the numbers:

$$
7 + 2 + 17 + 10 = 36
$$

Divide by the number of terms (4):

$$
\frac{36}{4} = 9
$$

The mean is indeed 9, confirming the solution is correct.

---

### Final Answer

$$
\boxed{17}
$$

Solution: If those 4 numbers average to 9, they must sum to $4\times 9 = 36$.  Then, we simply subtract the other three numbers from 36: $36 - 7 - 2 - 10 = \boxed{17} = x$.


Explanation:
Okay, so I need to find the equation of a line given its parametric equations and express it in the form y = mx + b. The parametric equations are:

$$
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 8 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 3 \end{pmatrix}
$$

Hmm, let me recall how parametric equations work. A parametric line is usually given by a point on the line plus a direction vector multiplied by a parameter t. In this case, the point is (8, -1) and the direction vector is (2, 3). So, this means that when t = 0, we're at the point (8, -1), and as t increases, we move along the direction of the vector (2, 3).

To convert this into the slope-intercept form y = mx + b, I need to find the slope m and the y-intercept b. Let me think about how to do that.

First, the direction vector (2, 3) gives me the slope of the line. The slope is rise over run, which is the change in y divided by the change in x. Since the direction vector is (2, 3), that means for every 2 units we move in the x-direction, we move 3 units in the y-direction. Therefore, the slope m should be 3/2. Wait, is that right? Let me verify. Yes, because slope is Δy/Δx, so 3 divided by 2. So m = 3/2. Got that part.

Now, I need to find b, the y-intercept. To do this, I can use the point-slope form of a line once I have the slope. But since I have a specific point on the line, (8, -1), I can plug that into the equation y = mx + b and solve for b.

So let's write that out. If the slope is 3/2, then the equation is y = (3/2)x + b. Now plug in x = 8 and y = -1:

-1 = (3/2)(8) + b

Let me compute (3/2)(8). That would be 3 * 4 = 12. So,

-1 = 12 + b

To solve for b, subtract 12 from both sides:

b = -1 - 12 = -13

Wait, so then the equation would be y = (3/2)x - 13? Let me check if this is correct. Let me verify using another point on the line.

For example, when t = 1, the point on the line is (8 + 2*1, -1 + 3*1) = (10, 2). Let me plug x = 10 into my equation y = (3/2)x -13. Then y should be (3/2)(10) -13 = 15 -13 = 2. Which matches the y-coordinate. Good.

Another check: when t = 0, the point is (8, -1). Plugging x = 8 into the equation: y = (3/2)(8) -13 = 12 -13 = -1. Correct.

So that seems to check out. Therefore, the ordered pair (m, b) is (3/2, -13). But let me think again if there's another way to approach this, just to be sure.

Alternatively, since parametric equations are given, I can express x and y in terms of t and eliminate t to find the Cartesian equation.

From the parametric equations:

x = 8 + 2t

y = -1 + 3t

So, let me solve the first equation for t:

x = 8 + 2t => x - 8 = 2t => t = (x - 8)/2

Now plug this into the equation for y:

y = -1 + 3*((x - 8)/2)

Simplify:

y = -1 + (3/2)(x - 8)

Multiply out the terms:

y = -1 + (3/2)x - (3/2)*8

Compute (3/2)*8: that's 12. So:

y = -1 + (3/2)x - 12

Combine the constants:

-1 -12 = -13

Thus, y = (3/2)x -13. Same result as before. So that's consistent.

Therefore, the slope m is 3/2 and the y-intercept b is -13. Therefore, the ordered pair is (3/2, -13).

But let me make sure that I didn't make any calculation errors here. Let me check my algebra again.

Starting with solving for t:

x = 8 + 2t. So subtract 8: x - 8 = 2t. Divide by 2: t = (x -8)/2. Correct.

Then plug into y:

y = -1 + 3t = -1 + 3*(x -8)/2. Yes. Then expanding:

Multiply 3/2 by x: (3/2)x. Then 3/2 times -8: which is -24/2 = -12. So y = -1 -12 + (3/2)x. Then -1 -12 is -13. So yes, y = (3/2)x -13. Perfect.

Alternatively, another way to check if the line is correct is to compute the direction vector from the slope-intercept form. The direction vector should be parallel to (2, 3). In slope-intercept form, the slope is 3/2, which corresponds to a direction vector of (2, 3), as we have. So that's consistent. Because if you take the slope as rise over run, 3/2, then a direction vector would be (2, 3). So that's correct.

Alternatively, if we wanted to find a direction vector from the equation y = (3/2)x -13, the direction vector could be (2, 3) as we saw, since the slope is 3/2. So that's all consistent.

Therefore, I think I can be confident that the answer is m = 3/2 and b = -13.

Wait, just to make sure, let me check if there's any chance I mixed up the direction vector components. The direction vector is (2,3), which is (Δx, Δy). So slope is Δy / Δx = 3/2. So that's correct. So m is indeed 3/2. If the direction vector was (3,2), then slope would be 2/3, but in this case, it's (2,3), so 3/2. So correct.

Another check: suppose we take t = -1. Then the point would be (8 + 2*(-1), -1 + 3*(-1)) = (8 -2, -1 -3) = (6, -4). Let's plug x = 6 into the equation y = (3/2)x -13: y = (3/2)*6 -13 = 9 -13 = -4. Which matches. Perfect.

Another check: t = 0.5. Then x = 8 + 2*(0.5) = 8 + 1 = 9. y = -1 + 3*(0.5) = -1 + 1.5 = 0.5. Plugging into the equation: y = (3/2)*9 -13 = 13.5 -13 = 0.5. Correct again.

So all these checks are working. Therefore, I can be confident that (m, b) is (3/2, -13).

**Final Answer**
The ordered pair is \boxed{\left( \dfrac{3}{2}, -13 \right)}.

To find the equation of a line in slope-intercept form $ y = mx + b $, we are given its parametric form:

$$
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 8 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 3 \end{pmatrix}
$$

---

### Step 1: Identify the slope $ m $

The direction vector $ \begin{pmatrix} 2 \\ 3 \end{pmatrix} $ tells us how the line changes as the parameter $ t $ varies. The slope $ m $ of the line is defined as the ratio of the change in $ y $ to the change in $ x $, which corresponds to:

$$
m = \frac{\Delta y}{\Delta x} = \frac{3}{2}
$$

---

### Step 2: Use point-slope form to find the y-intercept $ b $

We know a point on the line: $ (x, y) = (8, -1) $.

Using the point-slope form of a line:

$$
y - y_0 = m(x - x_0)
$$

Substituting $ m = \frac{3}{2} $, $ x_0 = 8 $, and $ y_0 = -1 $:

$$
y + 1 = \frac{3}{2}(x - 8)
$$

Now, simplify to slope-intercept form $ y = mx + b $:

$$
y + 1 = \frac{3}{2}x - 12 \\
y = \frac{3}{2}x - 12 - 1 \\
y = \frac{3}{2}x - 13
$$

---

### Step 3: Confirm the result

We can double-check using the parametric expressions directly:

From the parametric form:

$$
x = 8 + 2t \quad \text{and} \quad y = -1 + 3t
$$

Solve for $ t $ in terms of $ x $:

$$
x = 8 + 2t \Rightarrow t = \frac{x - 8}{2}
$$

Substitute into the equation for $ y $:

$$
y = -1 + 3\left(\frac{x - 8}{2}\right) \\
y = -1 + \frac{3(x - 8)}{2} = -1 + \frac{3x}{2} - 12 \\
y = \frac{3}{2}x - 13
$$

This confirms that the slope $ m = \frac{3}{2} $ and the y-intercept $ b = -13 $.

---

### Final Answer

$$
\boxed{\left( \dfrac{3}{2}, -13 \right)}
$$

[Assistant]
Explanation:
Okay, so I need to find the equation of a line given its parametric equations and express it in the form y = mx + b. The parametric equations are:

$$
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 8 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 3 \end{pmatrix}
$$

Hmm, let me recall how parametric equations work. A parametric line is usually given by a point on the line plus a direction vector multiplied by a parameter t. In this case, the point is (8, -1) and the direction vector is (2, 3). So, this means that when t = 0, we're at the point (8, -1), and as t increases, we move along the direction of the vector (2, 3).

To convert this into the slope-intercept form y = mx + b, I need to find the slope m and the y-intercept b. Let me think about how to do that.

First, the direction vector (2, 3) gives me the slope of the line. The slope is rise over run, which is the change in y divided by the change in x. Since the direction vector is (2, 3), that means for every 2 units we move in the x-direction, we move 3 units in the y-direction. Therefore, the slope m should be 3/2. Wait, is that right? Let me verify. Yes, because slope is Δy/Δx, so 3 divided by 2. So m = 3/2. Got that part.

Now, I need to find b, the y-intercept. To do this, I can use the point-slope form of a line once I have the slope. But since I have a specific point on the line, (8, -1), I can plug that into the equation y = mx + b and solve for b.

So let's write that out. If the slope is 3/2, then the equation is y = (3/2)x + b. Now plug in x = 8 and y = -1:

-1 = (3/2)(8) + b

Let me compute (3/2)(8). That would be 3 * 4 = 12. So,

-1 = 12 + b

To solve for b, subtract 12 from both sides:

b = -1 - 12 = -13

Wait, so then the equation would be y = (3/2)x - 13? Let me check if this is correct. Let me verify using another point on the line.

For example, when t = 1, the point on the line is (8 + 2*1, -1 + 3*1) = (10, 2). Let me plug x = 10 into my equation y = (3/2)x -13. Then y should be (3/2)(10) -13 = 15 -13 = 2. Which matches the y-coordinate. Good.

Another check: when t = 0, the point is (8, -1). Plugging x = 8 into the equation: y = (3/2)(8) -13 = 12 -13 = -1. Correct.

So that seems to check out. Therefore, the ordered pair (m, b) is (3/2, -13). But let me think again if there's another way to approach this, just to be sure.

Alternatively, since parametric equations are given, I can express x and y in terms of t and eliminate t to find the Cartesian equation.

From the parametric equations:

x = 8 + 2t

y = -1 + 3t

So, let me solve the first equation for t:

x = 8 + 2t => x - 8 = 2t => t = (x - 8)/2

Now plug this into the equation for y:

y = -1 + 3*((x - 8)/2)

Simplify:

y = -1 + (3/2)(x - 8)

Multiply out the terms:

y = -1 + (3/2)x - (3/2)*8

Compute (3/2)*8: that's 12. So:

y = -1 + (3/2)x - 12

Combine the constants:

-1 -12 = -13

Thus, y = (3/2)x -13. Same result as before. So that's consistent.

Therefore, the slope m is 3/2 and the y-intercept b is -13. Therefore, the ordered pair is (3/2, -13).

But let me make sure that I didn't make any calculation errors here. Let me check my algebra again.

Starting with solving for t:

x = 8 + 2t. So subtract 8: x - 8 = 2t. Divide by 2: t = (x -8)/2. Correct.

Then plug into y:

y = -1 + 3t = -1 + 3*(x -8)/2. Yes. Then expanding:

Multiply 3/2 by x: (3/2)x. Then 3/2 times -8: which is -24/2 = -12. So y = -1 -12 + (3/2)x. Then -1 -12 is -13. So yes, y = (3/2)x -13. Perfect.

Alternatively, another way to check if the line is correct is to compute the direction vector from the slope-intercept form. The direction vector should be parallel to (2, 3). In slope-intercept form, the slope is 3/2, which corresponds to a direction vector of (2, 3), as we have. So that's consistent. Because if you take the slope as rise over run, 3/2, then a direction vector would be (2, 3). So that's correct.

Alternatively, if we wanted to find a direction vector from the equation y = (3/2)x -13, the direction vector could be (2, 3) as we saw, since the slope is 3/2. So that's all consistent.

Therefore, I think I can be confident that the answer is m = 3/2 and b = -13.

Wait, just to make sure, let me check if there's any chance I mixed up the direction vector components. The direction vector is (2,3), which is (Δx, Δy). So slope is Δy / Δx = 3/2. So that's correct. So m is indeed 3/2. If the direction vector was (3,2), then slope would be 2/3, but in this case, it's (2,3), so 3/2. So correct.

Another check: suppose we take t = -1. Then the point would be (8 + 2*(-1), -1 + 3*(-1)) = (8 -2, -1 -3) = (6, -4). Let's plug x = 6 into the equation y = (3/2)x -13: y = (3/2)*6 -13 = 9 -13 = -4. Which matches. Perfect.

Another check: t = 0.5. Then x = 8 + 2*(0.5) = 8 + 1 = 9. y = -1 + 3*(0.5) = -1 + 1.5 = 0.5. Plugging into the equation: y = (3/2)*9 -13 = 13.5 -13 = 0.5. Correct again.

So all these checks are working. Therefore, I can be confident that (m, b) is (3/2, -13).

**Final Answer**
The ordered pair is \boxed{\left( \dfrac{3}{2}, -13 \right)}.

To find the equation of a line in slope-intercept form $ y = mx + b $, we are given its parametric form:

$$
\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 8 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 3 \end{pmatrix}
$$

---

### Step 1: Identify the slope $ m $

The direction vector $ \begin{pmatrix} 2 \\ 3 \end{pmatrix} $ tells us how the line changes as the parameter $ t $ varies. The slope $ m $ of the line is defined as the ratio of the change in $ y $ to the change in $ x $, which corresponds to:

$$
m = \frac{\Delta y}{\Delta x} = \frac{3}{2}
$$

---

### Step 2: Use point-slope form to find the y-intercept $ b $

We know a point on the line: $ (x, y) = (8, -1) $.

Using the point-slope form of a line:

$$
y - y_0 = m(x - x_0)
$$

Substituting $ m = \frac{3}{2} $, $ x_0 = 8 $, and $ y_0 = -1 $:

$$
y + 1 = \frac{3}{2}(x - 8)
$$

Now, simplify to slope-intercept form $ y = mx + b $:

$$
y + 1 = \frac{3}{2}x - 12 \\
y = \frac{3}{2}x - 12 - 1 \\
y = \frac{3}{2}x - 13
$$

---

### Step 3: Confirm the result

We can double-check using the parametric expressions directly:

From the parametric form:

$$
x = 8 + 2t \quad \text{and} \quad y = -1 + 3t
$$

Solve for $ t $ in terms of $ x $:

$$
x = 8 + 2t \Rightarrow t = \frac{x - 8}{2}
$$

Substitute into the equation for $ y $:

$$
y = -1 + 3\left(\frac{x - 8}{2}\right) \\
y = -1 + \frac{3(x - 8)}{2} = -1 + \frac{3x}{2} - 12 \\
y = \frac{3}{2}x - 13
$$

This confirms that the slope $ m = \frac{3}{2} $ and the y-intercept $ b = -13 $.

---

### Final Answer

$$
\boxed{\left( \dfrac{3}{2}, -13 \right)}
$$

Solution:
We have that $x = 8 + 2t$ and $y = -1 + 3t.$  Isolating $t$ in $x = 8 + 2t,$ we find
\[t = \frac{x - 8}{2}.\]Then
\begin{align*}
y &= -1 + 3t \\
&= -1 + 3 \cdot \frac{x - 8}{2} \\
&= \frac{3}{2} x - 13.
\end{align*}Thus, $(m,b) = \boxed{\left( \frac{3}{2}, -13 \right)}.$