Explanation:
Okay, so I need to find the area of a triangle with sides 5, 5, and 6 units. Hmm, let me think. First, since two sides are equal (both 5 units), this must be an isosceles triangle. I remember there are different ways to calculate the area of a triangle. The most common one is the base times height divided by two, right? But I don't know the height here. Maybe I can figure it out?

Alternatively, there's Heron's formula, which uses the lengths of all three sides. Let me recall how that works. Heron's formula says that if a triangle has sides of length a, b, and c, then the area is the square root of [s(s - a)(s - b)(s - c)], where s is the semi-perimeter. Let me try that.

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, which is 8. Okay, so s = 8.

Now plugging into Heron's formula: Area = sqrt[s(s - a)(s - b)(s - c)]. Let me substitute the values. 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 (since b is also 5)

s - c = 8 - 6 = 2

So multiplying all together: 8 * 3 * 3 * 2. Let me compute that step by step. 8*3 is 24, then 24*3 is 72, then 72*2 is 144. So the area is sqrt(144). The square root of 144 is 12. So the area is 12 square units? Wait, that seems straightforward. Let me check if I did that right.

Alternatively, maybe I can use the base and height method. Let me try that to verify. Let's take the base as 6 units. Then, since it's an isosceles triangle, the height can be found by splitting the base into two equal parts. So the base is 6, so each half is 3 units. Then, the height forms a right triangle with the side of 5 units and the base of 3 units. So using Pythagoras theorem: height squared + 3 squared = 5 squared. So height squared = 25 - 9 = 16. Therefore, height is 4 units. Then area is (base * height)/2 = (6 * 4)/2 = 24/2 = 12. Okay, same answer. So that confirms it. So the area is 12 square units.

Wait, so both methods give me 12. That seems solid. Let me just think if there's another way to check. Maybe using coordinates? Let me try that.

Let me place the triangle on a coordinate system. Let me put the base of 6 units along the x-axis, from (0, 0) to (6, 0). The third vertex is somewhere above the base. Since it's an isosceles triangle with sides 5, 5, 6, the third vertex should be centered above the base. Wait, the base is 6 units, so the midpoint is at (3, 0). The height we found earlier is 4 units, so the third vertex is at (3, 4). So the coordinates are (0, 0), (6, 0), and (3, 4). Let me compute the area using coordinates. There's the shoelace formula, which is another way.

Shoelace formula: For coordinates (x1, y1), (x2, y2), (x3, y3), the area is |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|.

Plugging in the points: (0, 0), (6, 0), (3, 4).

So compute:

0*(0 - 4) + 6*(4 - 0) + 3*(0 - 0) all over 2, absolute value.

Calculates to 0 + 6*4 + 3*0 = 24. Then divide by 2: 24/2 = 12. Absolute value is still 12. So same answer again. Okay, so that's three different methods all giving me 12. That seems pretty confident.

Alternatively, maybe I can use vectors or trigonometry? Let me try trigonometry. If I can find an angle, then use the formula (1/2)*ab*sin(theta). Let me see. Let me consider the two sides of 5 units and the included angle. Wait, but I don't know the angle. But maybe I can find it using the Law of Cosines?

Law of Cosines says that c² = a² + b² - 2ab*cos(theta). Let me take the sides 5, 5, and 6. Let me call the angle between the two sides of 5 units as theta. Then the side opposite theta is 6. So:

6² = 5² + 5² - 2*5*5*cos(theta)

So 36 = 25 + 25 - 50*cos(theta)

Simplify: 36 = 50 - 50*cos(theta)

Subtract 50: 36 - 50 = -50*cos(theta)

-14 = -50*cos(theta)

Divide both sides by -50: 14/50 = cos(theta)

Simplify: 7/25 = cos(theta)

Then sin(theta) can be found using sin²(theta) + cos²(theta) = 1.

So sin(theta) = sqrt(1 - (7/25)²) = sqrt(1 - 49/625) = sqrt(576/625) = 24/25.

Therefore, area is (1/2)*5*5*(24/25) = (1/2)*25*(24/25) = (1/2)*24 = 12. Again, same answer. Wow, so that's four different methods: Heron's formula, base-height, coordinate geometry, and trigonometry with Law of Cosines. All leading to 12. So I think that's pretty conclusive.

But just to recap, since I might have made a mistake in calculation somewhere. Let me check Heron's formula again. s = 8, so 8*(8-5)*(8-5)*(8-6) = 8*3*3*2. 8*3 is 24, 24*3 is 72, 72*2 is 144. Square root of 144 is 12. Correct. The base-height: base 6, height 4, area 12. Correct. Coordinates: shoelace formula gives 12. Correct. Law of Cosines: angle calculation leads to sin(theta) = 24/25, so area is 12. Correct. So all methods confirm the area is 12.

Therefore, I can be confident that the area is 12 square units.

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

To find the area of a triangle with sides measuring 5, 5, and 6 units, we can use multiple methods to confirm the result. Here's a step-by-step breakdown using **Heron's formula**, which is particularly well-suited for triangles with known side lengths.

---

### Step 1: Calculate the Semi-Perimeter

The semi-perimeter $ s $ of a triangle is given by:

$$
s = \frac{a + b + c}{2}
$$

Substituting the side lengths:

$$
s = \frac{5 + 5 + 6}{2} = \frac{16}{2} = 8
$$

---

### Step 2: Apply Heron's Formula

Heron's formula for the area $ A $ of a triangle is:

$$
A = \sqrt{s(s - a)(s - b)(s - c)}
$$

Substitute the known values:

$$
A = \sqrt{8(8 - 5)(8 - 5)(8 - 6)} = \sqrt{8 \cdot 3 \cdot 3 \cdot 2}
$$

$$
A = \sqrt{144} = 12
$$

---

### Step 3: Verification Using Base and Height

Since the triangle is isosceles (two sides of 5 units), we can drop a perpendicular from the vertex opposite the base (6 units) to the base, splitting it into two equal segments of 3 units each. This forms two right triangles with:

- Hypotenuse = 5 units
- One leg = 3 units
- The other leg = height $ h $

Using the Pythagorean Theorem:

$$
h^2 + 3^2 = 5^2 \Rightarrow h^2 = 25 - 9 = 16 \Rightarrow h = 4
$$

Now, the area is:

$$
A = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 6 \cdot 4 = 12
$$

---

### Step 4: Verification Using Coordinate Geometry (Shoelace Formula)

Place the triangle on the coordinate plane with vertices at:

- $ A = (0, 0) $
- $ B = (6, 0) $
- $ C = (3, 4) $

Using the shoelace formula:

$$
A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|
$$

$$
A = \frac{1}{2} |0(0 - 4) + 6(4 - 0) + 3(0 - 0)| = \frac{1}{2} |0 + 24 + 0| = \frac{1}{2} \cdot 24 = 12
$$

---

### Step 5: Verification Using Trigonometry (Law of Cosines)

Let the angle between the two sides of length 5 be $ \theta $. Using the Law of Cosines:

$$
6^2 = 5^2 + 5^2 - 2 \cdot 5 \cdot 5 \cdot \cos(\theta)
$$

$$
36 = 25 + 25 - 50 \cos(\theta) \Rightarrow 36 = 50 - 50 \cos(\theta)
$$

$$
\cos(\theta) = \frac{14}{50} = \frac{7}{25}
$$

Then:

$$
\sin(\theta) = \sqrt{1 - \left(\frac{7}{25}\right)^2} = \sqrt{1 - \frac{49}{625}} = \sqrt{\frac{576}{625}} = \frac{24}{25}
$$

Now, the area is:

$$
A = \frac{1}{2} \cdot 5 \cdot 5 \cdot \sin(\theta) = \frac{1}{2} \cdot 25 \cdot \frac{24}{25} = \frac{1}{2} \cdot 24 = 12
$$

---

### Final Answer

All methods consistently 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. First, since two sides are equal (both 5 units), this must be an isosceles triangle. I remember there are different ways to calculate the area of a triangle. The most common one is the base times height divided by two, right? But I don't know the height here. Maybe I can figure it out?

Alternatively, there's Heron's formula, which uses the lengths of all three sides. Let me recall how that works. Heron's formula says that if a triangle has sides of length a, b, and c, then the area is the square root of [s(s - a)(s - b)(s - c)], where s is the semi-perimeter. Let me try that.

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, which is 8. Okay, so s = 8.

Now plugging into Heron's formula: Area = sqrt[s(s - a)(s - b)(s - c)]. Let me substitute the values. 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 (since b is also 5)

s - c = 8 - 6 = 2

So multiplying all together: 8 * 3 * 3 * 2. Let me compute that step by step. 8*3 is 24, then 24*3 is 72, then 72*2 is 144. So the area is sqrt(144). The square root of 144 is 12. So the area is 12 square units? Wait, that seems straightforward. Let me check if I did that right.

Alternatively, maybe I can use the base and height method. Let me try that to verify. Let's take the base as 6 units. Then, since it's an isosceles triangle, the height can be found by splitting the base into two equal parts. So the base is 6, so each half is 3 units. Then, the height forms a right triangle with the side of 5 units and the base of 3 units. So using Pythagoras theorem: height squared + 3 squared = 5 squared. So height squared = 25 - 9 = 16. Therefore, height is 4 units. Then area is (base * height)/2 = (6 * 4)/2 = 24/2 = 12. Okay, same answer. So that confirms it. So the area is 12 square units.

Wait, so both methods give me 12. That seems solid. Let me just think if there's another way to check. Maybe using coordinates? Let me try that.

Let me place the triangle on a coordinate system. Let me put the base of 6 units along the x-axis, from (0, 0) to (6, 0). The third vertex is somewhere above the base. Since it's an isosceles triangle with sides 5, 5, 6, the third vertex should be centered above the base. Wait, the base is 6 units, so the midpoint is at (3, 0). The height we found earlier is 4 units, so the third vertex is at (3, 4). So the coordinates are (0, 0), (6, 0), and (3, 4). Let me compute the area using coordinates. There's the shoelace formula, which is another way.

Shoelace formula: For coordinates (x1, y1), (x2, y2), (x3, y3), the area is |(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))/2|.

Plugging in the points: (0, 0), (6, 0), (3, 4).

So compute:

0*(0 - 4) + 6*(4 - 0) + 3*(0 - 0) all over 2, absolute value.

Calculates to 0 + 6*4 + 3*0 = 24. Then divide by 2: 24/2 = 12. Absolute value is still 12. So same answer again. Okay, so that's three different methods all giving me 12. That seems pretty confident.

Alternatively, maybe I can use vectors or trigonometry? Let me try trigonometry. If I can find an angle, then use the formula (1/2)*ab*sin(theta). Let me see. Let me consider the two sides of 5 units and the included angle. Wait, but I don't know the angle. But maybe I can find it using the Law of Cosines?

Law of Cosines says that c² = a² + b² - 2ab*cos(theta). Let me take the sides 5, 5, and 6. Let me call the angle between the two sides of 5 units as theta. Then the side opposite theta is 6. So:

6² = 5² + 5² - 2*5*5*cos(theta)

So 36 = 25 + 25 - 50*cos(theta)

Simplify: 36 = 50 - 50*cos(theta)

Subtract 50: 36 - 50 = -50*cos(theta)

-14 = -50*cos(theta)

Divide both sides by -50: 14/50 = cos(theta)

Simplify: 7/25 = cos(theta)

Then sin(theta) can be found using sin²(theta) + cos²(theta) = 1.

So sin(theta) = sqrt(1 - (7/25)²) = sqrt(1 - 49/625) = sqrt(576/625) = 24/25.

Therefore, area is (1/2)*5*5*(24/25) = (1/2)*25*(24/25) = (1/2)*24 = 12. Again, same answer. Wow, so that's four different methods: Heron's formula, base-height, coordinate geometry, and trigonometry with Law of Cosines. All leading to 12. So I think that's pretty conclusive.

But just to recap, since I might have made a mistake in calculation somewhere. Let me check Heron's formula again. s = 8, so 8*(8-5)*(8-5)*(8-6) = 8*3*3*2. 8*3 is 24, 24*3 is 72, 72*2 is 144. Square root of 144 is 12. Correct. The base-height: base 6, height 4, area 12. Correct. Coordinates: shoelace formula gives 12. Correct. Law of Cosines: angle calculation leads to sin(theta) = 24/25, so area is 12. Correct. So all methods confirm the area is 12.

Therefore, I can be confident that the area is 12 square units.

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

To find the area of a triangle with sides measuring 5, 5, and 6 units, we can use multiple methods to confirm the result. Here's a step-by-step breakdown using **Heron's formula**, which is particularly well-suited for triangles with known side lengths.

---

### Step 1: Calculate the Semi-Perimeter

The semi-perimeter $ s $ of a triangle is given by:

$$
s = \frac{a + b + c}{2}
$$

Substituting the side lengths:

$$
s = \frac{5 + 5 + 6}{2} = \frac{16}{2} = 8
$$

---

### Step 2: Apply Heron's Formula

Heron's formula for the area $ A $ of a triangle is:

$$
A = \sqrt{s(s - a)(s - b)(s - c)}
$$

Substitute the known values:

$$
A = \sqrt{8(8 - 5)(8 - 5)(8 - 6)} = \sqrt{8 \cdot 3 \cdot 3 \cdot 2}
$$

$$
A = \sqrt{144} = 12
$$

---

### Step 3: Verification Using Base and Height

Since the triangle is isosceles (two sides of 5 units), we can drop a perpendicular from the vertex opposite the base (6 units) to the base, splitting it into two equal segments of 3 units each. This forms two right triangles with:

- Hypotenuse = 5 units
- One leg = 3 units
- The other leg = height $ h $

Using the Pythagorean Theorem:

$$
h^2 + 3^2 = 5^2 \Rightarrow h^2 = 25 - 9 = 16 \Rightarrow h = 4
$$

Now, the area is:

$$
A = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot 6 \cdot 4 = 12
$$

---

### Step 4: Verification Using Coordinate Geometry (Shoelace Formula)

Place the triangle on the coordinate plane with vertices at:

- $ A = (0, 0) $
- $ B = (6, 0) $
- $ C = (3, 4) $

Using the shoelace formula:

$$
A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|
$$

$$
A = \frac{1}{2} |0(0 - 4) + 6(4 - 0) + 3(0 - 0)| = \frac{1}{2} |0 + 24 + 0| = \frac{1}{2} \cdot 24 = 12
$$

---

### Step 5: Verification Using Trigonometry (Law of Cosines)

Let the angle between the two sides of length 5 be $ \theta $. Using the Law of Cosines:

$$
6^2 = 5^2 + 5^2 - 2 \cdot 5 \cdot 5 \cdot \cos(\theta)
$$

$$
36 = 25 + 25 - 50 \cos(\theta) \Rightarrow 36 = 50 - 50 \cos(\theta)
$$

$$
\cos(\theta) = \frac{14}{50} = \frac{7}{25}
$$

Then:

$$
\sin(\theta) = \sqrt{1 - \left(\frac{7}{25}\right)^2} = \sqrt{1 - \frac{49}{625}} = \sqrt{\frac{576}{625}} = \frac{24}{25}
$$

Now, the area is:

$$
A = \frac{1}{2} \cdot 5 \cdot 5 \cdot \sin(\theta) = \frac{1}{2} \cdot 25 \cdot \frac{24}{25} = \frac{1}{2} \cdot 24 = 12
$$

---

### Final Answer

All methods consistently 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, I know that m and n are integers, so they can be positive, negative, or zero. The equation is linear, so there are probably multiple solutions. My goal is to find the pair (m, n) that makes |m - n| as small as possible.

Maybe I can express one variable in terms of the other and then try to minimize the absolute difference. Let me try solving for m in terms of n.

Starting with 3m + 4n = 100. If I subtract 4n from both sides, I get 3m = 100 - 4n. Then, dividing both sides by 3, m = (100 - 4n)/3.

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

100 - 4n ≡ 0 mod 3
Which simplifies to:
-4n ≡ -100 mod 3
But maybe it's easier to compute 100 mod 3 and 4 mod 3.

Let me recall that 3*33 = 99, so 100 ≡ 1 mod 3. Similarly, 4 ≡ 1 mod 3. So substituting back:

1 - (1)n ≡ 0 mod 3
Which simplifies to:
1 - n ≡ 0 mod 3
Therefore, -n ≡ -1 mod 3
Which means n ≡ 1 mod 3.

So n must be congruent to 1 modulo 3. That is, n can be written as n = 3k + 1, where k is an integer.

So substituting back into the equation for m:

m = (100 - 4n)/3
But n = 3k + 1, so:

m = [100 - 4*(3k + 1)] / 3
= [100 - 12k - 4]/3
= [96 - 12k]/3
= 32 - 4k

So m = 32 - 4k and n = 3k + 1.

Therefore, m and n can be expressed in terms of integer k. 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.

So, I need to find integer k that makes 7k as close as possible to 31.

Let me solve 7k ≈ 31.

Dividing both sides by 7: k ≈ 31/7 ≈ 4.42857...

So the closest integers to 4.42857 are 4 and 5. Let me check both.

First, k = 4:

|31 - 7*4| = |31 - 28| = |3| = 3

k = 5:

|31 - 7*5| = |31 - 35| = | -4 | = 4

So between k=4 and k=5, the minimum is 3.

But wait, maybe there are other k's that could give a smaller value? Let me check k=3 and k=6 just to be sure.

k=3:

|31 - 21| = |10| = 10

k=6:

|31 - 42| = | -11 | = 11

So those are larger. How about k=4.428...? But k has to be integer, so 4 and 5 are the closest. So the minimal value is 3?

Wait, but let me check if there are other possible k's. For example, maybe negative k?

Let me check k=2:

|31 - 14| = 17

k=1:

|31 - 7| = 24

k=0:

|31 - 0| = 31

k=-1:

|31 - (-7)| = |38| = 38

So as k decreases, the value increases. Similarly, as k increases beyond 5, it increases. So the minimal value is indeed 3 when k=4.

But let me check the corresponding m and n for k=4 to make sure that they satisfy the original equation.

For k=4:

n = 3*4 + 1 = 13

m = 32 - 4*4 = 32 - 16 = 16

Check 3m + 4n = 3*16 + 4*13 = 48 + 52 = 100. Correct.

So |m - n| = |16 - 13| = 3.

Is there a possibility of getting a smaller value? Let me think.

Suppose there is another solution where |m - n| is less than 3, like 2 or 1. Let me check if that's possible.

Suppose |m - n| = 1. Then m - n = 1 or m - n = -1.

Case 1: m - n = 1. Then m = n + 1. Substitute into the equation:

3(n + 1) + 4n = 100
3n + 3 + 4n = 100
7n + 3 = 100
7n = 97
n = 97/7 ≈ 13.857... Not integer.

Case 2: m - n = -1. Then m = n - 1. Substitute:

3(n - 1) + 4n = 100
3n - 3 + 4n = 100
7n - 3 = 100
7n = 103
n = 103/7 ≈ 14.714... Not integer.

So no solutions for |m - n| = 1.

How about |m - n| = 2?

Case 1: m - n = 2. Then m = n + 2. Substitute:

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

Case 2: m - n = -2. Then m = n - 2. Substitute:

3(n - 2) + 4n = 100
3n - 6 + 4n = 100
7n - 6 = 100
7n = 106
n = 106/7 ≈ 15.142... Not integer.

So no solutions for |m - n| = 2.

Therefore, the minimal possible value is 3.

But wait, let me check if there are other solutions where |m - n| could be smaller. Maybe I missed something?

Alternatively, maybe there's another way to approach this problem. Let me think.

Another approach: Let me consider the original equation 3m + 4n = 100. Let me think of m and n as integers.

We can think of this as a Diophantine equation. The general solution can be found once we have a particular solution.

But since I already found a particular solution when k=4, which gives m=16, n=13, and the difference is 3.

But maybe there are other solutions with smaller differences.

Alternatively, perhaps I can parametrize the solutions.

Since the equation is 3m + 4n = 100, the general solution can be written as follows.

First, find one particular solution. Let me find m and n such that 3m + 4n = 100.

For example, if n=1, then 3m + 4 = 100 => 3m = 96 => m=32. So (m, n) = (32, 1) is a solution.

Then, the general solution can be found by adding multiples of the coefficients' coefficients. Wait, the general solution for ax + by = c is given by x = x0 + (b/d)*t, y = y0 - (a/d)*t, where d = gcd(a, b).

In this case, a=3, b=4, so d = gcd(3,4)=1. Therefore, the general solution is:

m = 32 + 4t
n = 1 - 3t

for any integer t.

Wait, but earlier I had m = 32 - 4k and n = 3k + 1. Let me check if these are equivalent.

If I set t = -k, then m = 32 + 4*(-k) = 32 - 4k, and n = 1 - 3*(-k) = 1 + 3k. Yes, same as before. So that's consistent.

Therefore, the general solution is m = 32 - 4t, n = 1 + 3t, where t is integer.

So then, m - n = (32 - 4t) - (1 + 3t) = 31 - 7t.

Therefore, |m - n| = |31 - 7t|.

So again, we need to find integer t such that |31 - 7t| is minimized.

Which is the same as before. So solving 7t ≈ 31, t ≈ 4.428... So t=4 or t=5.

For t=4: |31 - 28| = 3.

For t=5: |31 - 35| = 4.

So again, the minimal is 3.

Therefore, the minimal value is 3.

But let me check if there are other solutions with different t that could give smaller differences.

For example, t=4 gives m=32 - 16=16, n=1 + 12=13.

t=3: m=32 - 12=20, n=1 + 9=10. Then |20 - 10|=10.

t=5: m=32 - 20=12, n=1 + 15=16. |12 - 16|=4.

t=2: m=32 - 8=24, n=1 + 6=7. |24 - 7|=17.

t=1: m=32 - 4=28, n=1 + 3=4. |28 - 4|=24.

t=0: m=32, n=1. |32 - 1|=31.

t=-1: m=32 + 4=36, n=1 - 3= -2. |36 - (-2)|=38.

So yeah, as t moves away from 4, the difference increases.

So the minimal is indeed 3.

But let me check another particular solution. For example, if I take another particular solution.

Suppose I take n= -2, then 3m + 4*(-2) = 100 => 3m - 8 = 100 => 3m = 108 => m=36. So (36, -2).

Then, using the general solution, m=36 - 4t, n=-2 + 3t.

Wait, but according to the previous general solution, if we take t=-1, we get m=32 -4*(-1)=32 +4=36, n=1 +3*(-1)=1 -3=-2. So same as above.

So regardless of the particular solution, the general solution is parametrized by t, and the difference |m -n| is |31 -7t|.

So, same conclusion.

Alternatively, maybe there's another way to think about this problem.

Suppose we want to minimize |m - n|. Let me denote d = m - n. Then, m = n + d.

Substituting 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.

So 100 - 3d ≡ 0 mod 7
Which implies 3d ≡ 100 mod 7
Compute 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.

What is the inverse of 3 mod 7?

Looking for x such that 3x ≡ 1 mod 7. Testing x=5: 3*5=15≡1 mod7. Yes.

Therefore, multiplying both sides by 5:

d ≡ 2*5 mod7
d ≡ 10 mod7
d ≡ 3 mod7.

Therefore, d ≡ 3 mod7.

So d can be written as d = 7k + 3, where k is integer.

But d = m - n, and we want to minimize |d|.

So the possible values of d are ..., -11, -4, 3, 10, 17,...

So the minimal |d| is 3, which occurs when k=0.

Therefore, the minimal |m -n| is 3.

This confirms the previous result.

Therefore, regardless of the approach, the minimal value is 3.

But just to be thorough, let me check if there are any other possible d's with smaller absolute values.

Since d ≡ 3 mod7, the closest possible values to zero are d=3 and d=-4.

Because if d=-4, then |-4|=4, which is larger than 3.

So the minimal is indeed 3.

Therefore, the answer is 3.

But let me check with d=3.

If d=3, then n=(100 - 3*3)/7=(100 -9)/7=91/7=13. So n=13, then m = n + d =13 +3=16. Which is the same solution as before.

So yes, that works.

Alternatively, if d=-4, then n=(100 -3*(-4))/7=(100 +12)/7=112/7=16. Then m = n + d =16 + (-4)=12. Which is the solution when t=5 earlier.

So |m -n|=4.

Therefore, the minimal is indeed 3.

Therefore, I think I can be confident that the smallest possible value of |m -n| is 3.

**Final Answer**
The smallest possible value of $|m - n|$ is \boxed{3}.

To solve the problem, we are given the equation:

$$
3m + 4n = 100
$$

and we are to find the **smallest possible value** of $ |m - n| $, where $ m $ and $ n $ are integers.

---

### Step 1: Express One Variable in Terms of the Other

We can solve for $ m $ in terms of $ n $:

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

For $ m $ to be an integer, the numerator $ 100 - 4n $ must be divisible by 3. That is:

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

We simplify this congruence:

- $ 100 \equiv 1 \pmod{3} $
- $ 4 \equiv 1 \pmod{3} $

So:

$$
1 - n \equiv 0 \pmod{3} \Rightarrow n \equiv 1 \pmod{3}
$$

Thus, $ n $ must be of the form:

$$
n = 3k + 1 \quad \text{for some integer } k
$$

Substituting back into the expression for $ m $:

$$
m = \frac{100 - 4(3k + 1)}{3} = \frac{100 - 12k - 4}{3} = \frac{96 - 12k}{3} = 32 - 4k
$$

So we now have:

$$
m = 32 - 4k, \quad n = 3k + 1
$$

---

### Step 2: Minimize $ |m - n| $

We compute:

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

So:

$$
|m - n| = |31 - 7k|
$$

We now seek the integer $ k $ that minimizes $ |31 - 7k| $.

---

### Step 3: Find the Closest Integer to $ \frac{31}{7} $

$$
\frac{31}{7} \approx 4.4286
$$

So the closest integers are $ k = 4 $ and $ k = 5 $.

- For $ k = 4 $: $ |31 - 7 \cdot 4| = |31 - 28| = 3 $
- For $ k = 5 $: $ |31 - 7 \cdot 5| = |31 - 35| = 4 $

Thus, the minimum value is $ 3 $, achieved when $ k = 4 $.

---

### Step 4: Verify the Solution

For $ k = 4 $:

- $ n = 3k + 1 = 13 $
- $ m = 32 - 4k = 16 $

Check the original equation:

$$
3m + 4n = 3(16) + 4(13) = 48 + 52 = 100
$$

Correct.

Compute $ |m - n| = |16 - 13| = 3 $

---

### Step 5: Confirm Minimality

We can also consider the general form of the solution:

$$
m = 32 - 4t, \quad n = 1 + 3t
$$

Then:

$$
m - n = 31 - 7t \Rightarrow |m - n| = |31 - 7t|
$$

We again find that the minimum occurs at $ t = 4 $, giving $ |m - n| = 3 $.

Trying other values of $ t $ (e.g., $ t = 3, 5, 2, 6 $) results in larger values of $ |m - n| $, confirming that 3 is the smallest possible value.

---

### 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, I know that m and n are integers, so they can be positive, negative, or zero. The equation is linear, so there are probably multiple solutions. My goal is to find the pair (m, n) that makes |m - n| as small as possible.

Maybe I can express one variable in terms of the other and then try to minimize the absolute difference. Let me try solving for m in terms of n.

Starting with 3m + 4n = 100. If I subtract 4n from both sides, I get 3m = 100 - 4n. Then, dividing both sides by 3, m = (100 - 4n)/3.

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

100 - 4n ≡ 0 mod 3
Which simplifies to:
-4n ≡ -100 mod 3
But maybe it's easier to compute 100 mod 3 and 4 mod 3.

Let me recall that 3*33 = 99, so 100 ≡ 1 mod 3. Similarly, 4 ≡ 1 mod 3. So substituting back:

1 - (1)n ≡ 0 mod 3
Which simplifies to:
1 - n ≡ 0 mod 3
Therefore, -n ≡ -1 mod 3
Which means n ≡ 1 mod 3.

So n must be congruent to 1 modulo 3. That is, n can be written as n = 3k + 1, where k is an integer.

So substituting back into the equation for m:

m = (100 - 4n)/3
But n = 3k + 1, so:

m = [100 - 4*(3k + 1)] / 3
= [100 - 12k - 4]/3
= [96 - 12k]/3
= 32 - 4k

So m = 32 - 4k and n = 3k + 1.

Therefore, m and n can be expressed in terms of integer k. 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.

So, I need to find integer k that makes 7k as close as possible to 31.

Let me solve 7k ≈ 31.

Dividing both sides by 7: k ≈ 31/7 ≈ 4.42857...

So the closest integers to 4.42857 are 4 and 5. Let me check both.

First, k = 4:

|31 - 7*4| = |31 - 28| = |3| = 3

k = 5:

|31 - 7*5| = |31 - 35| = | -4 | = 4

So between k=4 and k=5, the minimum is 3.

But wait, maybe there are other k's that could give a smaller value? Let me check k=3 and k=6 just to be sure.

k=3:

|31 - 21| = |10| = 10

k=6:

|31 - 42| = | -11 | = 11

So those are larger. How about k=4.428...? But k has to be integer, so 4 and 5 are the closest. So the minimal value is 3?

Wait, but let me check if there are other possible k's. For example, maybe negative k?

Let me check k=2:

|31 - 14| = 17

k=1:

|31 - 7| = 24

k=0:

|31 - 0| = 31

k=-1:

|31 - (-7)| = |38| = 38

So as k decreases, the value increases. Similarly, as k increases beyond 5, it increases. So the minimal value is indeed 3 when k=4.

But let me check the corresponding m and n for k=4 to make sure that they satisfy the original equation.

For k=4:

n = 3*4 + 1 = 13

m = 32 - 4*4 = 32 - 16 = 16

Check 3m + 4n = 3*16 + 4*13 = 48 + 52 = 100. Correct.

So |m - n| = |16 - 13| = 3.

Is there a possibility of getting a smaller value? Let me think.

Suppose there is another solution where |m - n| is less than 3, like 2 or 1. Let me check if that's possible.

Suppose |m - n| = 1. Then m - n = 1 or m - n = -1.

Case 1: m - n = 1. Then m = n + 1. Substitute into the equation:

3(n + 1) + 4n = 100
3n + 3 + 4n = 100
7n + 3 = 100
7n = 97
n = 97/7 ≈ 13.857... Not integer.

Case 2: m - n = -1. Then m = n - 1. Substitute:

3(n - 1) + 4n = 100
3n - 3 + 4n = 100
7n - 3 = 100
7n = 103
n = 103/7 ≈ 14.714... Not integer.

So no solutions for |m - n| = 1.

How about |m - n| = 2?

Case 1: m - n = 2. Then m = n + 2. Substitute:

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

Case 2: m - n = -2. Then m = n - 2. Substitute:

3(n - 2) + 4n = 100
3n - 6 + 4n = 100
7n - 6 = 100
7n = 106
n = 106/7 ≈ 15.142... Not integer.

So no solutions for |m - n| = 2.

Therefore, the minimal possible value is 3.

But wait, let me check if there are other solutions where |m - n| could be smaller. Maybe I missed something?

Alternatively, maybe there's another way to approach this problem. Let me think.

Another approach: Let me consider the original equation 3m + 4n = 100. Let me think of m and n as integers.

We can think of this as a Diophantine equation. The general solution can be found once we have a particular solution.

But since I already found a particular solution when k=4, which gives m=16, n=13, and the difference is 3.

But maybe there are other solutions with smaller differences.

Alternatively, perhaps I can parametrize the solutions.

Since the equation is 3m + 4n = 100, the general solution can be written as follows.

First, find one particular solution. Let me find m and n such that 3m + 4n = 100.

For example, if n=1, then 3m + 4 = 100 => 3m = 96 => m=32. So (m, n) = (32, 1) is a solution.

Then, the general solution can be found by adding multiples of the coefficients' coefficients. Wait, the general solution for ax + by = c is given by x = x0 + (b/d)*t, y = y0 - (a/d)*t, where d = gcd(a, b).

In this case, a=3, b=4, so d = gcd(3,4)=1. Therefore, the general solution is:

m = 32 + 4t
n = 1 - 3t

for any integer t.

Wait, but earlier I had m = 32 - 4k and n = 3k + 1. Let me check if these are equivalent.

If I set t = -k, then m = 32 + 4*(-k) = 32 - 4k, and n = 1 - 3*(-k) = 1 + 3k. Yes, same as before. So that's consistent.

Therefore, the general solution is m = 32 - 4t, n = 1 + 3t, where t is integer.

So then, m - n = (32 - 4t) - (1 + 3t) = 31 - 7t.

Therefore, |m - n| = |31 - 7t|.

So again, we need to find integer t such that |31 - 7t| is minimized.

Which is the same as before. So solving 7t ≈ 31, t ≈ 4.428... So t=4 or t=5.

For t=4: |31 - 28| = 3.

For t=5: |31 - 35| = 4.

So again, the minimal is 3.

Therefore, the minimal value is 3.

But let me check if there are other solutions with different t that could give smaller differences.

For example, t=4 gives m=32 - 16=16, n=1 + 12=13.

t=3: m=32 - 12=20, n=1 + 9=10. Then |20 - 10|=10.

t=5: m=32 - 20=12, n=1 + 15=16. |12 - 16|=4.

t=2: m=32 - 8=24, n=1 + 6=7. |24 - 7|=17.

t=1: m=32 - 4=28, n=1 + 3=4. |28 - 4|=24.

t=0: m=32, n=1. |32 - 1|=31.

t=-1: m=32 + 4=36, n=1 - 3= -2. |36 - (-2)|=38.

So yeah, as t moves away from 4, the difference increases.

So the minimal is indeed 3.

But let me check another particular solution. For example, if I take another particular solution.

Suppose I take n= -2, then 3m + 4*(-2) = 100 => 3m - 8 = 100 => 3m = 108 => m=36. So (36, -2).

Then, using the general solution, m=36 - 4t, n=-2 + 3t.

Wait, but according to the previous general solution, if we take t=-1, we get m=32 -4*(-1)=32 +4=36, n=1 +3*(-1)=1 -3=-2. So same as above.

So regardless of the particular solution, the general solution is parametrized by t, and the difference |m -n| is |31 -7t|.

So, same conclusion.

Alternatively, maybe there's another way to think about this problem.

Suppose we want to minimize |m - n|. Let me denote d = m - n. Then, m = n + d.

Substituting 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.

So 100 - 3d ≡ 0 mod 7
Which implies 3d ≡ 100 mod 7
Compute 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.

What is the inverse of 3 mod 7?

Looking for x such that 3x ≡ 1 mod 7. Testing x=5: 3*5=15≡1 mod7. Yes.

Therefore, multiplying both sides by 5:

d ≡ 2*5 mod7
d ≡ 10 mod7
d ≡ 3 mod7.

Therefore, d ≡ 3 mod7.

So d can be written as d = 7k + 3, where k is integer.

But d = m - n, and we want to minimize |d|.

So the possible values of d are ..., -11, -4, 3, 10, 17,...

So the minimal |d| is 3, which occurs when k=0.

Therefore, the minimal |m -n| is 3.

This confirms the previous result.

Therefore, regardless of the approach, the minimal value is 3.

But just to be thorough, let me check if there are any other possible d's with smaller absolute values.

Since d ≡ 3 mod7, the closest possible values to zero are d=3 and d=-4.

Because if d=-4, then |-4|=4, which is larger than 3.

So the minimal is indeed 3.

Therefore, the answer is 3.

But let me check with d=3.

If d=3, then n=(100 - 3*3)/7=(100 -9)/7=91/7=13. So n=13, then m = n + d =13 +3=16. Which is the same solution as before.

So yes, that works.

Alternatively, if d=-4, then n=(100 -3*(-4))/7=(100 +12)/7=112/7=16. Then m = n + d =16 + (-4)=12. Which is the solution when t=5 earlier.

So |m -n|=4.

Therefore, the minimal is indeed 3.

Therefore, I think I can be confident that the smallest possible value of |m -n| is 3.

**Final Answer**
The smallest possible value of $|m - n|$ is \boxed{3}.

To solve the problem, we are given the equation:

$$
3m + 4n = 100
$$

and we are to find the **smallest possible value** of $ |m - n| $, where $ m $ and $ n $ are integers.

---

### Step 1: Express One Variable in Terms of the Other

We can solve for $ m $ in terms of $ n $:

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

For $ m $ to be an integer, the numerator $ 100 - 4n $ must be divisible by 3. That is:

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

We simplify this congruence:

- $ 100 \equiv 1 \pmod{3} $
- $ 4 \equiv 1 \pmod{3} $

So:

$$
1 - n \equiv 0 \pmod{3} \Rightarrow n \equiv 1 \pmod{3}
$$

Thus, $ n $ must be of the form:

$$
n = 3k + 1 \quad \text{for some integer } k
$$

Substituting back into the expression for $ m $:

$$
m = \frac{100 - 4(3k + 1)}{3} = \frac{100 - 12k - 4}{3} = \frac{96 - 12k}{3} = 32 - 4k
$$

So we now have:

$$
m = 32 - 4k, \quad n = 3k + 1
$$

---

### Step 2: Minimize $ |m - n| $

We compute:

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

So:

$$
|m - n| = |31 - 7k|
$$

We now seek the integer $ k $ that minimizes $ |31 - 7k| $.

---

### Step 3: Find the Closest Integer to $ \frac{31}{7} $

$$
\frac{31}{7} \approx 4.4286
$$

So the closest integers are $ k = 4 $ and $ k = 5 $.

- For $ k = 4 $: $ |31 - 7 \cdot 4| = |31 - 28| = 3 $
- For $ k = 5 $: $ |31 - 7 \cdot 5| = |31 - 35| = 4 $

Thus, the minimum value is $ 3 $, achieved when $ k = 4 $.

---

### Step 4: Verify the Solution

For $ k = 4 $:

- $ n = 3k + 1 = 13 $
- $ m = 32 - 4k = 16 $

Check the original equation:

$$
3m + 4n = 3(16) + 4(13) = 48 + 52 = 100
$$

Correct.

Compute $ |m - n| = |16 - 13| = 3 $

---

### Step 5: Confirm Minimality

We can also consider the general form of the solution:

$$
m = 32 - 4t, \quad n = 1 + 3t
$$

Then:

$$
m - n = 31 - 7t \Rightarrow |m - n| = |31 - 7t|
$$

We again find that the minimum occurs at $ t = 4 $, giving $ |m - n| = 3 $.

Trying other values of $ t $ (e.g., $ t = 3, 5, 2, 6 $) results in larger values of $ |m - n| $, confirming that 3 is the smallest possible value.

---

### 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, so I need to find the positive square root of the product 10 times 15 times 24. Let me think about how to approach this. First, maybe I should calculate the product of those three numbers and then take the square root. But before jumping into multiplying them all together, maybe there's a smarter way to simplify things? Let me see.

Let me write down the numbers: 10, 15, and 24. Hmm, maybe I can factor each of them into primes? That might help me see if there are any perfect squares in there that I can take out of the square root. Let me try that.

Starting with 10. The prime factors of 10 are 2 and 5. So, 10 = 2 × 5.

Next, 15. The prime factors of 15 are 3 and 5. So, 15 = 3 × 5.

Then, 24. Let me break that down. 24 divided by 2 is 12, divided by 2 again is 6, and again by 2 is 3. So, 24 = 2³ × 3. Alternatively, 24 = 2 × 2 × 2 × 3.

So, putting it all together, the product 10 × 15 × 24 would be:

(2 × 5) × (3 × 5) × (2³ × 3)

Let me multiply all those together. Let me combine like terms. For the prime factors:

- For 2: there's one from 10, and three from 24. So total of 1 + 3 = 4.
- For 3: one from 15 and one from 24. So total of 1 + 1 = 2.
- For 5: one from 10 and one from 15. So total of 1 + 1 = 2.

So altogether, the product is 2⁴ × 3² × 5². Let me check that again. Let me count:

Original factors:

10: 2^1 * 5^1

15: 3^1 * 5^1

24: 2^3 * 3^1

Multiplying them together:

2^(1+3) * 3^(1+1) * 5^(1+1) = 2^4 * 3^2 * 5^2. Yes, that's correct.

So the product is 2⁴ × 3² × 5². Now, the question is asking for the positive square root of this product. So, sqrt(2⁴ × 3² × 5²).

To take the square root, I can take the square root of each factor separately. Remember that sqrt(a^b) = a^(b/2). So, let me apply that.

sqrt(2⁴) = 2^(4/2) = 2² = 4

sqrt(3²) = 3^(2/2) = 3^1 = 3

sqrt(5²) = 5^(2/2) = 5^1 = 5

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

First, 4 × 3 is 12, then 12 × 5 is 60. So, the square root of the product is 60.

Wait, let me verify that. Maybe I should check by calculating the product first and then taking the square root. Let me do that as a cross-check.

Calculating 10 × 15 × 24. Let me compute step by step.

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

Breaking down 150 × 24. 150 × 20 = 3000, and 150 × 4 = 600. So, 3000 + 600 = 3600. Therefore, the product is 3600. Then, the square root of 3600 is... Well, 60 × 60 is 3600, so sqrt(3600) is 60.

Okay, that matches my previous result. So, that seems correct. Therefore, the positive square root is 60.

But let me think again if there's another way to approach this. Maybe by simplifying before multiplying? For example, maybe pairing numbers to make squares?

Looking back at the original numbers: 10, 15, 24. Let me see if I can pair them in a way that makes squares. For example, 10 and 15 have a common factor of 5. 10 is 2×5, 15 is 3×5. If I multiply them together, 10×15 = 2×5×3×5 = 2×3×5². Then, multiplying by 24, which is 2³×3. So, (2×3×5²) × (2³×3) = 2^(1+3) × 3^(1+1) × 5² = 2^4 × 3² × 5², which is the same as before. So, same result.

Alternatively, maybe I can take square roots step by step? For example, sqrt(10 × 15 × 24) = sqrt(10 × 15) × sqrt(24). Let me try that.

First, sqrt(10 × 15) = sqrt(150). Then sqrt(150) × sqrt(24). Let me compute sqrt(150) × sqrt(24). That's sqrt(150 × 24). Wait, that's going back to the original problem. So that doesn't help. Maybe another way?

Alternatively, sqrt(10 × 15 × 24) = sqrt( (10 × 24) × 15 ). Let me compute 10 × 24 first. 10 × 24 is 240. Then, sqrt(240 × 15). Let me compute 240 × 15. 240 × 10 = 2400, 240 × 5 = 1200, so total is 2400 + 1200 = 3600. Then sqrt(3600) is 60. Same result again.

Alternatively, maybe I can factor numbers to make squares. Let me think. For example, 10 × 15 × 24. Let me see if I can write this as something squared times something else.

Earlier, we saw that the product is 2^4 × 3^2 × 5^2. Which can be written as (2^2 × 3 × 5)^2. Let me check that. (2^2 × 3 × 5) is 4 × 3 × 5 = 60. Then, 60 squared is 3600. Which is the product. Therefore, sqrt(3600) is 60.

So, that's another way to see it. So, the square root of (something squared) is the something, so sqrt( (60)^2 ) = 60.

Therefore, regardless of the method, I get 60.

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

Original numbers:

10 = 2 × 5

15 = 3 × 5

24 = 2³ × 3

Multiplying them all together:

2 × 5 × 3 × 5 × 2³ × 3

Grouping the 2s: 2^1 × 2^3 = 2^(1+3) = 2^4

Grouping the 3s: 3^1 × 3^1 = 3^(1+1) = 3^2

Grouping the 5s: 5^1 × 5^1 = 5^(1+1) = 5^2

So, 2^4 × 3^2 × 5^2. Correct.

Therefore, square root is 2^(4/2) × 3^(2/2) × 5^(2/2) = 2² × 3 × 5 = 4 × 3 × 5 = 60.

Yes, that's correct.

Alternatively, maybe I can think of the original numbers as multiples of squares? Let me see. For example, 10 is 2×5, 15 is 3×5, 24 is 4×6. Wait, 24 is 4×6, and 4 is a square. So, sqrt(10 × 15 × 24) = sqrt(10 × 15 × 4 × 6) = sqrt(4 × 10 × 15 × 6). Then sqrt(4) is 2, so that becomes 2 × sqrt(10 × 15 × 6). Let me compute 10 × 15 × 6.

10 × 15 is 150, 150 × 6 is 900. Then sqrt(900) is 30. Therefore, total is 2 × 30 = 60.

Oh, that's another way to break it down. So, by recognizing that 24 is 4×6, which is a square times something else. Then, taking the square root of 4 first.

So, that's another method. So, that also gives me 60.

So, multiple methods all leading to the same answer. Therefore, I can be confident that the positive square root of the product 10 × 15 × 24 is 60.

But just to make sure, let me compute the product again step by step.

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

Well, 150 × 24 can be thought of as 150 × (20 + 4) = 150 × 20 + 150 × 4.

150 × 20: 150 × 2 × 10 = 300 × 10 = 3000.

150 × 4: 150 × 2 × 2 = 300 × 2 = 600.

Adding them together: 3000 + 600 = 3600.

So, the product is 3600. Then sqrt(3600).

What's sqrt(3600)? Well, sqrt(36 × 100) = sqrt(36) × sqrt(100) = 6 × 10 = 60.

Yes, that's another way to see it. So, breaking down 3600 into 36 × 100, both of which are perfect squares.

Therefore, sqrt(36 × 100) = 6 × 10 = 60.

So, all methods converge to 60.

Therefore, I think I can be confident that the answer is 60.

But just to make sure, let me check with another approach. Let me compute the square of 60 and see if it equals 10 × 15 × 24.

60 squared is 60 × 60. Let me compute that.

60 × 60 = 3600.

Earlier, we saw that 10 × 15 × 24 = 3600. Therefore, yes, 60 squared is 3600, so sqrt(3600) is 60.

Therefore, that's correct.

Alternatively, if I didn't remember that 60 squared is 3600, I could compute it step by step.

60 × 60:

Well, 6 × 6 = 36, and since there are two zeros, it's 36 followed by two zeros, so 3600.

Yes.

Therefore, all checks out.

So, after going through multiple methods—prime factorization, breaking down the product into known squares, step-by-step multiplication and square root, cross-verifying with different factor groupings—it seems that the answer is definitely 60.

I don't see any mistakes in my reasoning, and all the different approaches confirm the same result. Therefore, I can be confident that the positive square root of the product 10 × 15 × 24 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 can approach the problem by first simplifying the expression using prime factorization and then applying the properties of square roots.

---

### Step 1: Prime Factorization

We begin by breaking each number into its prime factors:

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

Now, multiply all these together:

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

Grouping like terms:

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

So the product becomes:

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

---

### Step 2: Apply the Square Root

We now take the square root of the entire expression:

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

Using the property $\sqrt{a^b} = a^{b/2}$, we simplify each term:

- $\sqrt{2^4} = 2^{4/2} = 2^2 = 4$
- $\sqrt{3^2} = 3^{2/2} = 3^1 = 3$
- $\sqrt{5^2} = 5^{2/2} = 5^1 = 5$

Now multiply the results:

$$
4 \times 3 \times 5 = 60
$$

---

### Step 3: Verification

To double-check, we can compute the product directly:

$$
10 \times 15 = 150,\quad 150 \times 24 = 3600
$$

Then:

$$
\sqrt{3600} = 60
$$

This confirms our earlier result.

---

### Final Answer

$$
\boxed{60}
$$

[Assistant]
Explanation:
Okay, so I need to find the positive square root of the product 10 times 15 times 24. Let me think about how to approach this. First, maybe I should calculate the product of those three numbers and then take the square root. But before jumping into multiplying them all together, maybe there's a smarter way to simplify things? Let me see.

Let me write down the numbers: 10, 15, and 24. Hmm, maybe I can factor each of them into primes? That might help me see if there are any perfect squares in there that I can take out of the square root. Let me try that.

Starting with 10. The prime factors of 10 are 2 and 5. So, 10 = 2 × 5.

Next, 15. The prime factors of 15 are 3 and 5. So, 15 = 3 × 5.

Then, 24. Let me break that down. 24 divided by 2 is 12, divided by 2 again is 6, and again by 2 is 3. So, 24 = 2³ × 3. Alternatively, 24 = 2 × 2 × 2 × 3.

So, putting it all together, the product 10 × 15 × 24 would be:

(2 × 5) × (3 × 5) × (2³ × 3)

Let me multiply all those together. Let me combine like terms. For the prime factors:

- For 2: there's one from 10, and three from 24. So total of 1 + 3 = 4.
- For 3: one from 15 and one from 24. So total of 1 + 1 = 2.
- For 5: one from 10 and one from 15. So total of 1 + 1 = 2.

So altogether, the product is 2⁴ × 3² × 5². Let me check that again. Let me count:

Original factors:

10: 2^1 * 5^1

15: 3^1 * 5^1

24: 2^3 * 3^1

Multiplying them together:

2^(1+3) * 3^(1+1) * 5^(1+1) = 2^4 * 3^2 * 5^2. Yes, that's correct.

So the product is 2⁴ × 3² × 5². Now, the question is asking for the positive square root of this product. So, sqrt(2⁴ × 3² × 5²).

To take the square root, I can take the square root of each factor separately. Remember that sqrt(a^b) = a^(b/2). So, let me apply that.

sqrt(2⁴) = 2^(4/2) = 2² = 4

sqrt(3²) = 3^(2/2) = 3^1 = 3

sqrt(5²) = 5^(2/2) = 5^1 = 5

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

First, 4 × 3 is 12, then 12 × 5 is 60. So, the square root of the product is 60.

Wait, let me verify that. Maybe I should check by calculating the product first and then taking the square root. Let me do that as a cross-check.

Calculating 10 × 15 × 24. Let me compute step by step.

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

Breaking down 150 × 24. 150 × 20 = 3000, and 150 × 4 = 600. So, 3000 + 600 = 3600. Therefore, the product is 3600. Then, the square root of 3600 is... Well, 60 × 60 is 3600, so sqrt(3600) is 60.

Okay, that matches my previous result. So, that seems correct. Therefore, the positive square root is 60.

But let me think again if there's another way to approach this. Maybe by simplifying before multiplying? For example, maybe pairing numbers to make squares?

Looking back at the original numbers: 10, 15, 24. Let me see if I can pair them in a way that makes squares. For example, 10 and 15 have a common factor of 5. 10 is 2×5, 15 is 3×5. If I multiply them together, 10×15 = 2×5×3×5 = 2×3×5². Then, multiplying by 24, which is 2³×3. So, (2×3×5²) × (2³×3) = 2^(1+3) × 3^(1+1) × 5² = 2^4 × 3² × 5², which is the same as before. So, same result.

Alternatively, maybe I can take square roots step by step? For example, sqrt(10 × 15 × 24) = sqrt(10 × 15) × sqrt(24). Let me try that.

First, sqrt(10 × 15) = sqrt(150). Then sqrt(150) × sqrt(24). Let me compute sqrt(150) × sqrt(24). That's sqrt(150 × 24). Wait, that's going back to the original problem. So that doesn't help. Maybe another way?

Alternatively, sqrt(10 × 15 × 24) = sqrt( (10 × 24) × 15 ). Let me compute 10 × 24 first. 10 × 24 is 240. Then, sqrt(240 × 15). Let me compute 240 × 15. 240 × 10 = 2400, 240 × 5 = 1200, so total is 2400 + 1200 = 3600. Then sqrt(3600) is 60. Same result again.

Alternatively, maybe I can factor numbers to make squares. Let me think. For example, 10 × 15 × 24. Let me see if I can write this as something squared times something else.

Earlier, we saw that the product is 2^4 × 3^2 × 5^2. Which can be written as (2^2 × 3 × 5)^2. Let me check that. (2^2 × 3 × 5) is 4 × 3 × 5 = 60. Then, 60 squared is 3600. Which is the product. Therefore, sqrt(3600) is 60.

So, that's another way to see it. So, the square root of (something squared) is the something, so sqrt( (60)^2 ) = 60.

Therefore, regardless of the method, I get 60.

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

Original numbers:

10 = 2 × 5

15 = 3 × 5

24 = 2³ × 3

Multiplying them all together:

2 × 5 × 3 × 5 × 2³ × 3

Grouping the 2s: 2^1 × 2^3 = 2^(1+3) = 2^4

Grouping the 3s: 3^1 × 3^1 = 3^(1+1) = 3^2

Grouping the 5s: 5^1 × 5^1 = 5^(1+1) = 5^2

So, 2^4 × 3^2 × 5^2. Correct.

Therefore, square root is 2^(4/2) × 3^(2/2) × 5^(2/2) = 2² × 3 × 5 = 4 × 3 × 5 = 60.

Yes, that's correct.

Alternatively, maybe I can think of the original numbers as multiples of squares? Let me see. For example, 10 is 2×5, 15 is 3×5, 24 is 4×6. Wait, 24 is 4×6, and 4 is a square. So, sqrt(10 × 15 × 24) = sqrt(10 × 15 × 4 × 6) = sqrt(4 × 10 × 15 × 6). Then sqrt(4) is 2, so that becomes 2 × sqrt(10 × 15 × 6). Let me compute 10 × 15 × 6.

10 × 15 is 150, 150 × 6 is 900. Then sqrt(900) is 30. Therefore, total is 2 × 30 = 60.

Oh, that's another way to break it down. So, by recognizing that 24 is 4×6, which is a square times something else. Then, taking the square root of 4 first.

So, that's another method. So, that also gives me 60.

So, multiple methods all leading to the same answer. Therefore, I can be confident that the positive square root of the product 10 × 15 × 24 is 60.

But just to make sure, let me compute the product again step by step.

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

Well, 150 × 24 can be thought of as 150 × (20 + 4) = 150 × 20 + 150 × 4.

150 × 20: 150 × 2 × 10 = 300 × 10 = 3000.

150 × 4: 150 × 2 × 2 = 300 × 2 = 600.

Adding them together: 3000 + 600 = 3600.

So, the product is 3600. Then sqrt(3600).

What's sqrt(3600)? Well, sqrt(36 × 100) = sqrt(36) × sqrt(100) = 6 × 10 = 60.

Yes, that's another way to see it. So, breaking down 3600 into 36 × 100, both of which are perfect squares.

Therefore, sqrt(36 × 100) = 6 × 10 = 60.

So, all methods converge to 60.

Therefore, I think I can be confident that the answer is 60.

But just to make sure, let me check with another approach. Let me compute the square of 60 and see if it equals 10 × 15 × 24.

60 squared is 60 × 60. Let me compute that.

60 × 60 = 3600.

Earlier, we saw that 10 × 15 × 24 = 3600. Therefore, yes, 60 squared is 3600, so sqrt(3600) is 60.

Therefore, that's correct.

Alternatively, if I didn't remember that 60 squared is 3600, I could compute it step by step.

60 × 60:

Well, 6 × 6 = 36, and since there are two zeros, it's 36 followed by two zeros, so 3600.

Yes.

Therefore, all checks out.

So, after going through multiple methods—prime factorization, breaking down the product into known squares, step-by-step multiplication and square root, cross-verifying with different factor groupings—it seems that the answer is definitely 60.

I don't see any mistakes in my reasoning, and all the different approaches confirm the same result. Therefore, I can be confident that the positive square root of the product 10 × 15 × 24 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 can approach the problem by first simplifying the expression using prime factorization and then applying the properties of square roots.

---

### Step 1: Prime Factorization

We begin by breaking each number into its prime factors:

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

Now, multiply all these together:

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

Grouping like terms:

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

So the product becomes:

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

---

### Step 2: Apply the Square Root

We now take the square root of the entire expression:

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

Using the property $\sqrt{a^b} = a^{b/2}$, we simplify each term:

- $\sqrt{2^4} = 2^{4/2} = 2^2 = 4$
- $\sqrt{3^2} = 3^{2/2} = 3^1 = 3$
- $\sqrt{5^2} = 5^{2/2} = 5^1 = 5$

Now multiply the results:

$$
4 \times 3 \times 5 = 60
$$

---

### Step 3: Verification

To double-check, we can compute the product directly:

$$
10 \times 15 = 150,\quad 150 \times 24 = 3600
$$

Then:

$$
\sqrt{3600} = 60
$$

This confirms our earlier 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, so I need to figure out how many students are at Baker Middle School. The problem says that two-thirds of the students take music, and there are 834 students who take music. Hmm, let me think about how to approach this.

First, let me restate the problem in my own words. If two-thirds of the total number of students take music, and that number is 834, then I need to find the total number of students. So, basically, if I let the total number of students be some number, let's call it x, then two-thirds of x is equal to 834. So, mathematically, that would be (2/3)x = 834. Right?

So, to find x, I need to solve this equation. How do I solve for x when it's multiplied by a fraction? Well, I think I need to do the opposite operation. Since it's multiplied by 2/3, I should divide both sides by 2/3 to isolate x. Dividing by a fraction is the same as multiplying by its reciprocal, so dividing by 2/3 is the same as multiplying by 3/2. Let me write that down:

(2/3)x = 834
x = 834 ÷ (2/3)
x = 834 × (3/2)

Okay, so now I need to compute 834 multiplied by 3/2. Let me do that step by step. Maybe I can break it down. Multiplying by 3/2 is the same as multiplying by 3 and then dividing by 2, or dividing by 2 and then multiplying by 3. Let me try both ways to see which is easier.

First way: Multiply 834 by 3 first. Let me calculate 834 × 3.

Well, 800 × 3 is 2400, and 34 × 3 is 102. So adding those together, 2400 + 102 = 2502. Then divide by 2. So 2502 ÷ 2. That would be 1251.

Alternatively, if I divide 834 by 2 first, then multiply by 3. Let me check that. 834 ÷ 2 is 417. Then 417 × 3. Let me compute that. 400 × 3 is 1200, and 17 × 3 is 51. Adding those together, 1200 + 51 = 1251. Same result. Okay, so that seems consistent.

So, x = 1251. Therefore, the total number of students at Baker Middle School is 1251. Let me just verify that to make sure I didn't make a mistake.

If there are 1251 students total, then two-thirds of them would be (2/3) × 1251. Let me compute that.

First, divide 1251 by 3. 1251 ÷ 3. Let me do that division. 3 goes into 12 four times, 4 × 3 is 12, subtract, bring down 5. 3 goes into 5 once, 1 × 3 is 3, subtract, bring down 1. 3 goes into 1 zero times, so 0, then bring down the 1. Wait, actually, maybe I should do this more carefully.

Wait, 3 × 400 is 1200. So 1251 - 1200 is 51. Then 51 ÷ 3 is 17. So total is 400 + 17 = 417. Then multiply by 2: 417 × 2 = 834. Which matches the number given in the problem. Perfect! So that checks out.

Therefore, my answer is 1251. Let me just make sure I didn't make any arithmetic errors. Let me recheck the multiplication steps.

Original equation: (2/3)x = 834. Solving for x: x = 834 × (3/2).

Calculating 834 × 3: 800 × 3 = 2400, 34 × 3 = 102. 2400 + 102 = 2502. Then 2502 ÷ 2 = 1251. Correct.

Alternatively, 834 ÷ 2 = 417, 417 × 3 = 1251. Correct.

And verifying by plugging back in: 1251 × 2/3 = 834. Correct.

So, all steps check out. Therefore, the total number of students is 1251.

I think that's it. I don't see any mistakes in my reasoning. The key was setting up the equation correctly and then performing the division and multiplication accurately. It's important to remember that dividing by a fraction is the same as multiplying by its reciprocal, which is a common point of confusion. But since I checked both ways of calculating 834 × 3/2 and got the same result, and then verified by plugging back into the original equation, I feel confident that 1251 is the correct answer.

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

To determine the total number of students at Baker Middle School, we are given that **two-thirds** of the students take music, and that **834 students** take music. Our goal is to find the total number of students, which we'll denote as $ x $.

---

### Step 1: Set Up the Equation

We are told that:

$$
\frac{2}{3} \text{ of the total students } = 834
$$

This can be written as:

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

---

### Step 2: Solve for $ x $

To isolate $ x $, we divide both sides of the equation by $ \frac{2}{3} $, which is the same as multiplying by its reciprocal:

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

Now, compute the multiplication:

$$
834 \times \frac{3}{2} = \frac{834 \times 3}{2} = \frac{2502}{2} = 1251
$$

---

### Step 3: Verify the Solution

To ensure the answer is correct, we can plug $ x = 1251 $ back into the original equation:

$$
\frac{2}{3} \times 1251 = \frac{2 \times 1251}{3} = \frac{2502}{3} = 834
$$

This matches the given number of students who take music, confirming our solution is accurate.

---

### Final Answer

$$
\boxed{1251}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many students are at Baker Middle School. The problem says that two-thirds of the students take music, and there are 834 students who take music. Hmm, let me think about how to approach this.

First, let me restate the problem in my own words. If two-thirds of the total number of students take music, and that number is 834, then I need to find the total number of students. So, basically, if I let the total number of students be some number, let's call it x, then two-thirds of x is equal to 834. So, mathematically, that would be (2/3)x = 834. Right?

So, to find x, I need to solve this equation. How do I solve for x when it's multiplied by a fraction? Well, I think I need to do the opposite operation. Since it's multiplied by 2/3, I should divide both sides by 2/3 to isolate x. Dividing by a fraction is the same as multiplying by its reciprocal, so dividing by 2/3 is the same as multiplying by 3/2. Let me write that down:

(2/3)x = 834
x = 834 ÷ (2/3)
x = 834 × (3/2)

Okay, so now I need to compute 834 multiplied by 3/2. Let me do that step by step. Maybe I can break it down. Multiplying by 3/2 is the same as multiplying by 3 and then dividing by 2, or dividing by 2 and then multiplying by 3. Let me try both ways to see which is easier.

First way: Multiply 834 by 3 first. Let me calculate 834 × 3.

Well, 800 × 3 is 2400, and 34 × 3 is 102. So adding those together, 2400 + 102 = 2502. Then divide by 2. So 2502 ÷ 2. That would be 1251.

Alternatively, if I divide 834 by 2 first, then multiply by 3. Let me check that. 834 ÷ 2 is 417. Then 417 × 3. Let me compute that. 400 × 3 is 1200, and 17 × 3 is 51. Adding those together, 1200 + 51 = 1251. Same result. Okay, so that seems consistent.

So, x = 1251. Therefore, the total number of students at Baker Middle School is 1251. Let me just verify that to make sure I didn't make a mistake.

If there are 1251 students total, then two-thirds of them would be (2/3) × 1251. Let me compute that.

First, divide 1251 by 3. 1251 ÷ 3. Let me do that division. 3 goes into 12 four times, 4 × 3 is 12, subtract, bring down 5. 3 goes into 5 once, 1 × 3 is 3, subtract, bring down 1. 3 goes into 1 zero times, so 0, then bring down the 1. Wait, actually, maybe I should do this more carefully.

Wait, 3 × 400 is 1200. So 1251 - 1200 is 51. Then 51 ÷ 3 is 17. So total is 400 + 17 = 417. Then multiply by 2: 417 × 2 = 834. Which matches the number given in the problem. Perfect! So that checks out.

Therefore, my answer is 1251. Let me just make sure I didn't make any arithmetic errors. Let me recheck the multiplication steps.

Original equation: (2/3)x = 834. Solving for x: x = 834 × (3/2).

Calculating 834 × 3: 800 × 3 = 2400, 34 × 3 = 102. 2400 + 102 = 2502. Then 2502 ÷ 2 = 1251. Correct.

Alternatively, 834 ÷ 2 = 417, 417 × 3 = 1251. Correct.

And verifying by plugging back in: 1251 × 2/3 = 834. Correct.

So, all steps check out. Therefore, the total number of students is 1251.

I think that's it. I don't see any mistakes in my reasoning. The key was setting up the equation correctly and then performing the division and multiplication accurately. It's important to remember that dividing by a fraction is the same as multiplying by its reciprocal, which is a common point of confusion. But since I checked both ways of calculating 834 × 3/2 and got the same result, and then verified by plugging back into the original equation, I feel confident that 1251 is the correct answer.

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

To determine the total number of students at Baker Middle School, we are given that **two-thirds** of the students take music, and that **834 students** take music. Our goal is to find the total number of students, which we'll denote as $ x $.

---

### Step 1: Set Up the Equation

We are told that:

$$
\frac{2}{3} \text{ of the total students } = 834
$$

This can be written as:

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

---

### Step 2: Solve for $ x $

To isolate $ x $, we divide both sides of the equation by $ \frac{2}{3} $, which is the same as multiplying by its reciprocal:

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

Now, compute the multiplication:

$$
834 \times \frac{3}{2} = \frac{834 \times 3}{2} = \frac{2502}{2} = 1251
$$

---

### Step 3: Verify the Solution

To ensure the answer is correct, we can plug $ x = 1251 $ back into the original equation:

$$
\frac{2}{3} \times 1251 = \frac{2 \times 1251}{3} = \frac{2502}{3} = 834
$$

This matches the given number of students who take music, confirming our solution is accurate.

---

### 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 is 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 plug in the x and y values of the point into the equation, it should hold true. In this case, the point is (a, b), which is (4, b) because a is given as 4. So I need to substitute x with 4 and y with b in the equation 3x + 2y = 12 and solve for b.

Let me write that down step by step. The original equation is 3x + 2y = 12. Substituting x = 4 and y = b gives me:

3*(4) + 2*(b) = 12.

Calculating 3 times 4, that's 12. So the equation becomes:

12 + 2b = 12.

Now, I need to solve for b. Let me subtract 12 from both sides to isolate the term with b. So:

12 + 2b - 12 = 12 - 12.

Simplifying both sides, the left side becomes 2b and the right side becomes 0. So:

2b = 0.

To solve for b, I divide both sides by 2:

b = 0 / 2.

Which simplifies to:

b = 0.

Wait, so is that it? Let me check my steps again to make sure I didn't make a mistake. Starting with substituting x = 4 into the equation:

3*4 is indeed 12. Then adding 2b equals 12. Subtract 12 from both sides gives 2b = 0, so b = 0. That seems straightforward. Maybe I can plug the values back into the original equation to verify.

Original equation: 3x + 2y = 12. If x = 4 and y = 0, then 3*4 + 2*0 = 12 + 0 = 12, which matches the right side of the equation. So that checks out. Therefore, b must be 0.

Hmm, but let me think again. Is there another way to approach this problem? Maybe by rearranging the equation to solve for y in terms of x first? Let me try that.

Starting with 3x + 2y = 12. If I solve for y, I can subtract 3x from both sides:

2y = 12 - 3x.

Then divide both sides by 2:

y = (12 - 3x)/2.

Simplify that:

y = 6 - (3/2)x.

So, if x is 4, then y would be 6 - (3/2)*4. Let me compute that. (3/2)*4 is 6, so 6 - 6 = 0. Yep, same result. So that confirms that when x is 4, y is 0. Therefore, b is 0.

I think that's solid. Both methods give me the same answer, and plugging it back into the original equation works. So I don't see any mistakes here. Maybe I can visualize it? Let me sketch the line mentally. The line 3x + 2y = 12. If x is 0, then 2y = 12, so y = 6. If y is 0, then 3x = 12, so x = 4. So the line crosses the x-axis at (4, 0) and the y-axis at (0, 6). Therefore, the point (4, 0) is on the line, which means when a = 4, b = 0. That makes sense. So that's another way to see it. The x-intercept is (4, 0), so when x is 4, y is 0. Therefore, b is 0.

I think that's thorough. All different approaches lead to the same conclusion. I don't think I made any errors here. The answer should be 0.

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

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

$$
3x + 2y = 12
$$

and we are told that $ a = 4 $, we proceed as follows:

---

### Step 1: Understand the condition

A point $ (a, b) $ lies on a line if its coordinates satisfy the equation of the line. That is, substituting $ x = a $ and $ y = b $ into the equation should make it true.

Given:
- $ a = 4 $
- The point is $ (4, b) $

Substitute $ x = 4 $ and $ y = b $ into the equation:

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

---

### Step 2: Simplify the equation

$$
12 + 2b = 12
$$

Subtract 12 from both sides:

$$
2b = 0
$$

Divide both sides by 2:

$$
b = 0
$$

---

### Step 3: Verify the solution

Plug $ x = 4 $ and $ y = 0 $ back into the original equation:

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

This confirms that the point $ (4, 0) $ lies on the line.

---

### Final Answer

$$
\boxed{0}
$$

[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 is 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 plug in the x and y values of the point into the equation, it should hold true. In this case, the point is (a, b), which is (4, b) because a is given as 4. So I need to substitute x with 4 and y with b in the equation 3x + 2y = 12 and solve for b.

Let me write that down step by step. The original equation is 3x + 2y = 12. Substituting x = 4 and y = b gives me:

3*(4) + 2*(b) = 12.

Calculating 3 times 4, that's 12. So the equation becomes:

12 + 2b = 12.

Now, I need to solve for b. Let me subtract 12 from both sides to isolate the term with b. So:

12 + 2b - 12 = 12 - 12.

Simplifying both sides, the left side becomes 2b and the right side becomes 0. So:

2b = 0.

To solve for b, I divide both sides by 2:

b = 0 / 2.

Which simplifies to:

b = 0.

Wait, so is that it? Let me check my steps again to make sure I didn't make a mistake. Starting with substituting x = 4 into the equation:

3*4 is indeed 12. Then adding 2b equals 12. Subtract 12 from both sides gives 2b = 0, so b = 0. That seems straightforward. Maybe I can plug the values back into the original equation to verify.

Original equation: 3x + 2y = 12. If x = 4 and y = 0, then 3*4 + 2*0 = 12 + 0 = 12, which matches the right side of the equation. So that checks out. Therefore, b must be 0.

Hmm, but let me think again. Is there another way to approach this problem? Maybe by rearranging the equation to solve for y in terms of x first? Let me try that.

Starting with 3x + 2y = 12. If I solve for y, I can subtract 3x from both sides:

2y = 12 - 3x.

Then divide both sides by 2:

y = (12 - 3x)/2.

Simplify that:

y = 6 - (3/2)x.

So, if x is 4, then y would be 6 - (3/2)*4. Let me compute that. (3/2)*4 is 6, so 6 - 6 = 0. Yep, same result. So that confirms that when x is 4, y is 0. Therefore, b is 0.

I think that's solid. Both methods give me the same answer, and plugging it back into the original equation works. So I don't see any mistakes here. Maybe I can visualize it? Let me sketch the line mentally. The line 3x + 2y = 12. If x is 0, then 2y = 12, so y = 6. If y is 0, then 3x = 12, so x = 4. So the line crosses the x-axis at (4, 0) and the y-axis at (0, 6). Therefore, the point (4, 0) is on the line, which means when a = 4, b = 0. That makes sense. So that's another way to see it. The x-intercept is (4, 0), so when x is 4, y is 0. Therefore, b is 0.

I think that's thorough. All different approaches lead to the same conclusion. I don't think I made any errors here. The answer should be 0.

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

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

$$
3x + 2y = 12
$$

and we are told that $ a = 4 $, we proceed as follows:

---

### Step 1: Understand the condition

A point $ (a, b) $ lies on a line if its coordinates satisfy the equation of the line. That is, substituting $ x = a $ and $ y = b $ into the equation should make it true.

Given:
- $ a = 4 $
- The point is $ (4, b) $

Substitute $ x = 4 $ and $ y = b $ into the equation:

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

---

### Step 2: Simplify the equation

$$
12 + 2b = 12
$$

Subtract 12 from both sides:

$$
2b = 0
$$

Divide both sides by 2:

$$
b = 0
$$

---

### Step 3: Verify the solution

Plug $ x = 4 $ and $ y = 0 $ back into the original equation:

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

This confirms that the point $ (4, 0) $ lies on the line.

---

### Final Answer

$$
\boxed{0}
$$

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 when the arithmetic mean of the numbers 7, 2, x, and 10 is 9. Let me think about how to approach this.

First, I remember that 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, I can set up an equation based on that definition.

Let me write that down. The formula for the arithmetic mean is:

Mean = (Sum of all numbers) / (Number of numbers)

Plugging in the values given:

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

Okay, so the equation is 9 equals the sum of those four numbers divided by 4. My goal is to solve for x.

First, maybe I should simplify the sum in the numerator. Let me add up the numbers I know. 7 plus 2 is 9, and 9 plus 10 is 19. So, the sum becomes 19 + x. Therefore, the equation simplifies to:

9 = (19 + x) / 4

Now, to solve for x, I need to get rid of the denominator. I can do that by multiplying both sides of the equation by 4. Let me do that step.

Multiplying both sides by 4:

9 * 4 = 19 + x

Calculating the left side: 9 times 4 is 36. So:

36 = 19 + x

Now, to isolate x, I need to subtract 19 from both sides of the equation. Let me do that.

36 - 19 = x

Calculating 36 minus 19... Hmm, 36 minus 20 would be 16, but since I subtracted 1 more than needed, I need to add 1 back. So, 16 + 1 = 17. Therefore, x equals 17.

Wait, let me check that again. 19 + x = 36. So, if I subtract 19 from both sides, x = 36 - 19. Let me compute 36 - 19.

Starting with 36 - 10 is 26, then subtract 9 more: 26 - 9 = 17. Yes, that's correct. So x is 17.

But let me verify my answer to make sure I didn't make a mistake. Let me plug x = 17 back into the original numbers and calculate the mean.

The numbers would be 7, 2, 17, and 10. Adding them up: 7 + 2 is 9, 9 + 17 is 26, and 26 + 10 is 36. Then divide by 4: 36 / 4 = 9. Which matches the given mean.

So that checks out. Therefore, x is indeed 17.

Wait, but let me go through the steps again just to be thorough. Sometimes, even if the check works, I might have made an error in setting up the equation.

Original problem: arithmetic mean of 7, 2, x, 10 is 9.

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

Adding the constants: 7 + 2 + 10 = 19. So, (19 + x)/4 = 9. Multiply both sides by 4: 19 + x = 36. Subtract 19: x = 17. Yep, same result.

Another way to think about it: the total sum should be mean multiplied by the number of terms. So, 9 * 4 = 36. The sum of the numbers is 36. The sum of the known numbers is 7 + 2 + 10 = 19. Therefore, the missing number is 36 - 19 = 17.

All methods lead to x = 17. I think that's solid.

Is there any other way this could be interpreted? For example, could the question be referring to something other than the arithmetic mean? But the problem specifically says "arithmetic mean," so that's straightforward.

Alternatively, could there be a typo in the problem? But assuming the problem is stated correctly, with numbers 7, 2, x, 10, and mean 9, then x is 17.

I don't see any other possible interpretations. So, I think the answer is 17.

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

To find the value of $ x $, we are given that the **arithmetic mean** of the numbers 7, 2, $ x $, and 10 is 9. The arithmetic mean is defined as the sum of the numbers divided by the number of terms.

---

### Step 1: Set up the equation

We know the formula for the arithmetic mean:

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

Substituting the known values:

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

---

### Step 2: Simplify the numerator

Add the known constants:

$$
7 + 2 + 10 = 19
$$

So the equation becomes:

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

---

### Step 3: Eliminate the denominator

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

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

$$
36 = 19 + x
$$

---

### Step 4: Solve for $ x $

Subtract 19 from both sides:

$$
x = 36 - 19
$$

$$
x = 17
$$

---

### Step 5: Verify the solution

Plug $ x = 17 $ back into the original set of numbers: 7, 2, 17, and 10.

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

Mean: $ \frac{36}{4} = 9 $, which matches the given mean.

---

### Final Answer

$$
\boxed{17}
$$

[Assistant]
Explanation:
Okay, so I need to find the value of x when the arithmetic mean of the numbers 7, 2, x, and 10 is 9. Let me think about how to approach this.

First, I remember that 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, I can set up an equation based on that definition.

Let me write that down. The formula for the arithmetic mean is:

Mean = (Sum of all numbers) / (Number of numbers)

Plugging in the values given:

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

Okay, so the equation is 9 equals the sum of those four numbers divided by 4. My goal is to solve for x.

First, maybe I should simplify the sum in the numerator. Let me add up the numbers I know. 7 plus 2 is 9, and 9 plus 10 is 19. So, the sum becomes 19 + x. Therefore, the equation simplifies to:

9 = (19 + x) / 4

Now, to solve for x, I need to get rid of the denominator. I can do that by multiplying both sides of the equation by 4. Let me do that step.

Multiplying both sides by 4:

9 * 4 = 19 + x

Calculating the left side: 9 times 4 is 36. So:

36 = 19 + x

Now, to isolate x, I need to subtract 19 from both sides of the equation. Let me do that.

36 - 19 = x

Calculating 36 minus 19... Hmm, 36 minus 20 would be 16, but since I subtracted 1 more than needed, I need to add 1 back. So, 16 + 1 = 17. Therefore, x equals 17.

Wait, let me check that again. 19 + x = 36. So, if I subtract 19 from both sides, x = 36 - 19. Let me compute 36 - 19.

Starting with 36 - 10 is 26, then subtract 9 more: 26 - 9 = 17. Yes, that's correct. So x is 17.

But let me verify my answer to make sure I didn't make a mistake. Let me plug x = 17 back into the original numbers and calculate the mean.

The numbers would be 7, 2, 17, and 10. Adding them up: 7 + 2 is 9, 9 + 17 is 26, and 26 + 10 is 36. Then divide by 4: 36 / 4 = 9. Which matches the given mean.

So that checks out. Therefore, x is indeed 17.

Wait, but let me go through the steps again just to be thorough. Sometimes, even if the check works, I might have made an error in setting up the equation.

Original problem: arithmetic mean of 7, 2, x, 10 is 9.

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

Adding the constants: 7 + 2 + 10 = 19. So, (19 + x)/4 = 9. Multiply both sides by 4: 19 + x = 36. Subtract 19: x = 17. Yep, same result.

Another way to think about it: the total sum should be mean multiplied by the number of terms. So, 9 * 4 = 36. The sum of the numbers is 36. The sum of the known numbers is 7 + 2 + 10 = 19. Therefore, the missing number is 36 - 19 = 17.

All methods lead to x = 17. I think that's solid.

Is there any other way this could be interpreted? For example, could the question be referring to something other than the arithmetic mean? But the problem specifically says "arithmetic mean," so that's straightforward.

Alternatively, could there be a typo in the problem? But assuming the problem is stated correctly, with numbers 7, 2, x, 10, and mean 9, then x is 17.

I don't see any other possible interpretations. So, I think the answer is 17.

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

To find the value of $ x $, we are given that the **arithmetic mean** of the numbers 7, 2, $ x $, and 10 is 9. The arithmetic mean is defined as the sum of the numbers divided by the number of terms.

---

### Step 1: Set up the equation

We know the formula for the arithmetic mean:

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

Substituting the known values:

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

---

### Step 2: Simplify the numerator

Add the known constants:

$$
7 + 2 + 10 = 19
$$

So the equation becomes:

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

---

### Step 3: Eliminate the denominator

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

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

$$
36 = 19 + x
$$

---

### Step 4: Solve for $ x $

Subtract 19 from both sides:

$$
x = 36 - 19
$$

$$
x = 17
$$

---

### Step 5: Verify the solution

Plug $ x = 17 $ back into the original set of numbers: 7, 2, 17, and 10.

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

Mean: $ \frac{36}{4} = 9 $, which matches the given mean.

---

### 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:

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

Alright, let me recall how parametric equations work. A parametric line is defined by a point on the line and a direction vector. In this case, the point is (8, -1) when t = 0, and the direction vector is <2, 3>. So, as t varies, the line is traced out by adding multiples of the direction vector to the initial point.

To convert this into the slope-intercept form y = mx + b, I need to find the slope (m) and the y-intercept (b).

First, let me think about the slope. The direction vector <2, 3> gives the direction of the line. The slope of a line is rise over run, which is the change in y over the change in x. In the direction vector, the run is 2 (the x-component) and the rise is 3 (the y-component). Therefore, the slope m should be 3/2. Let me confirm that. If I move 2 units in the x-direction, I move 3 units in the y-direction, so the slope is indeed 3/2. So m = 3/2. That seems straightforward.

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

Let me write that out. If x = 8 and y = -1, then:

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

Calculating (3/2)(8): 3/2 * 8 = 3 * 4 = 12. So:

-1 = 12 + b

Subtracting 12 from both sides:

b = -1 - 12 = -13

Wait, so b is -13? Let me check that again. If the line is y = (3/2)x - 13, does it pass through (8, -1)?

Let me plug x = 8 into the equation:

y = (3/2)(8) - 13 = 12 - 13 = -1. Yes, that works. So that seems correct.

But let me make sure I didn't make a mistake in the calculation. Let me go through it again.

Starting with the parametric equations:

x = 8 + 2t

y = -1 + 3t

So, if I solve for t in terms of x, I can substitute into the equation for y. Let me try that method as a check.

From the x equation:

x = 8 + 2t

Subtract 8: x - 8 = 2t

Divide by 2: t = (x - 8)/2

Now substitute this into the equation for y:

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

Simplify:

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

Distribute the 3/2:

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

Calculate (3/2)*8: 12, so:

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

Combine constants:

-1 - 12 = -13

So, y = (3/2)x - 13. Yep, same result. So that confirms that m = 3/2 and b = -13. Therefore, the ordered pair (m, b) is (3/2, -13).

But let me think again if there's another way to approach this. Maybe using the general form of a line?

Alternatively, the direction vector is <2, 3>, so the slope is 3/2 as before. Then, using point-slope form with the point (8, -1):

y - y1 = m(x - x1)

So:

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

Which simplifies to:

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

Which is:

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

Subtract 1:

y = (3/2)x - 12 - 1 = (3/2)x - 13

Same result again. So that's consistent. Therefore, I think my answer is correct.

But just to be thorough, let me check with another point on the line. Let's pick a value of t and see if the coordinates satisfy the equation y = (3/2)x - 13.

For example, let's take t = 1. Then:

x = 8 + 2(1) = 10

y = -1 + 3(1) = 2

Now plug x = 10 into the equation:

y = (3/2)(10) - 13 = 15 - 13 = 2. Perfect, that matches.

Another check: t = 0 gives (8, -1), which we already checked. How about t = -1?

x = 8 + 2(-1) = 8 - 2 = 6

y = -1 + 3(-1) = -1 - 3 = -4

Plug x = 6 into the equation:

y = (3/2)(6) - 13 = 9 - 13 = -4. Correct again.

So, all these checks confirm that the equation is correct. Therefore, the ordered pair (m, b) is (3/2, -13).

I think that's solid. I don't see any mistakes in my reasoning. All methods lead to the same result, and multiple points check out. So I'm confident that this is the correct answer.

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

To find the equation of the line in the form $ y = mx + b $, we start with the given parametric equations:

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

---

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

The direction vector of the line is $ \begin{pmatrix} 2 \\ 3 \end{pmatrix} $. The slope $ m $ of a line is the ratio of the change in $ y $ to the change in $ x $, which corresponds to the components of the direction vector:

$$
m = \frac{3}{2}
$$

---

### Step 2: Use a point on the line to find $ b $

We are given a point on the line when $ t = 0 $: $ (x, y) = (8, -1) $. Using the point-slope form of a line:

$$
y - y_1 = m(x - x_1)
$$

Substitute $ m = \frac{3}{2} $, $ x_1 = 8 $, and $ y_1 = -1 $:

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

Now simplify:

$$
y = \frac{3}{2}x - \frac{3}{2} \cdot 8 - 1 = \frac{3}{2}x - 12 - 1 = \frac{3}{2}x - 13
$$

---

### Step 3: Final Equation and Ordered Pair

The equation of the line is:

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

Thus, the ordered pair $ (m, b) $ is:

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

[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:

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

Alright, let me recall how parametric equations work. A parametric line is defined by a point on the line and a direction vector. In this case, the point is (8, -1) when t = 0, and the direction vector is <2, 3>. So, as t varies, the line is traced out by adding multiples of the direction vector to the initial point.

To convert this into the slope-intercept form y = mx + b, I need to find the slope (m) and the y-intercept (b).

First, let me think about the slope. The direction vector <2, 3> gives the direction of the line. The slope of a line is rise over run, which is the change in y over the change in x. In the direction vector, the run is 2 (the x-component) and the rise is 3 (the y-component). Therefore, the slope m should be 3/2. Let me confirm that. If I move 2 units in the x-direction, I move 3 units in the y-direction, so the slope is indeed 3/2. So m = 3/2. That seems straightforward.

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

Let me write that out. If x = 8 and y = -1, then:

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

Calculating (3/2)(8): 3/2 * 8 = 3 * 4 = 12. So:

-1 = 12 + b

Subtracting 12 from both sides:

b = -1 - 12 = -13

Wait, so b is -13? Let me check that again. If the line is y = (3/2)x - 13, does it pass through (8, -1)?

Let me plug x = 8 into the equation:

y = (3/2)(8) - 13 = 12 - 13 = -1. Yes, that works. So that seems correct.

But let me make sure I didn't make a mistake in the calculation. Let me go through it again.

Starting with the parametric equations:

x = 8 + 2t

y = -1 + 3t

So, if I solve for t in terms of x, I can substitute into the equation for y. Let me try that method as a check.

From the x equation:

x = 8 + 2t

Subtract 8: x - 8 = 2t

Divide by 2: t = (x - 8)/2

Now substitute this into the equation for y:

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

Simplify:

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

Distribute the 3/2:

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

Calculate (3/2)*8: 12, so:

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

Combine constants:

-1 - 12 = -13

So, y = (3/2)x - 13. Yep, same result. So that confirms that m = 3/2 and b = -13. Therefore, the ordered pair (m, b) is (3/2, -13).

But let me think again if there's another way to approach this. Maybe using the general form of a line?

Alternatively, the direction vector is <2, 3>, so the slope is 3/2 as before. Then, using point-slope form with the point (8, -1):

y - y1 = m(x - x1)

So:

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

Which simplifies to:

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

Which is:

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

Subtract 1:

y = (3/2)x - 12 - 1 = (3/2)x - 13

Same result again. So that's consistent. Therefore, I think my answer is correct.

But just to be thorough, let me check with another point on the line. Let's pick a value of t and see if the coordinates satisfy the equation y = (3/2)x - 13.

For example, let's take t = 1. Then:

x = 8 + 2(1) = 10

y = -1 + 3(1) = 2

Now plug x = 10 into the equation:

y = (3/2)(10) - 13 = 15 - 13 = 2. Perfect, that matches.

Another check: t = 0 gives (8, -1), which we already checked. How about t = -1?

x = 8 + 2(-1) = 8 - 2 = 6

y = -1 + 3(-1) = -1 - 3 = -4

Plug x = 6 into the equation:

y = (3/2)(6) - 13 = 9 - 13 = -4. Correct again.

So, all these checks confirm that the equation is correct. Therefore, the ordered pair (m, b) is (3/2, -13).

I think that's solid. I don't see any mistakes in my reasoning. All methods lead to the same result, and multiple points check out. So I'm confident that this is the correct answer.

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

To find the equation of the line in the form $ y = mx + b $, we start with the given parametric equations:

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

---

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

The direction vector of the line is $ \begin{pmatrix} 2 \\ 3 \end{pmatrix} $. The slope $ m $ of a line is the ratio of the change in $ y $ to the change in $ x $, which corresponds to the components of the direction vector:

$$
m = \frac{3}{2}
$$

---

### Step 2: Use a point on the line to find $ b $

We are given a point on the line when $ t = 0 $: $ (x, y) = (8, -1) $. Using the point-slope form of a line:

$$
y - y_1 = m(x - x_1)
$$

Substitute $ m = \frac{3}{2} $, $ x_1 = 8 $, and $ y_1 = -1 $:

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

Now simplify:

$$
y = \frac{3}{2}x - \frac{3}{2} \cdot 8 - 1 = \frac{3}{2}x - 12 - 1 = \frac{3}{2}x - 13
$$

---

### Step 3: Final Equation and Ordered Pair

The equation of the line is:

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

Thus, the ordered pair $ (m, b) $ is:

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

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, $(m,b) = \boxed{\left( \frac{3}{2}, -13 \right)}.$