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 can be written as 2 × 2 × 2 × 3, which is 2³ × 3.

So, putting it all together, the product 10 × 15 × 24 is equal to (2 × 5) × (3 × 5) × (2³ × 3). Let me write that out:

(2 × 5) × (3 × 5) × (2³ × 3) = 2 × 5 × 3 × 5 × 2³ × 3.

Now, let me combine like terms. Let me count the exponents for each prime factor.

For the prime number 2: there's one 2 from the first term and three 2s from the third term. So total exponents for 2: 1 + 3 = 4.

For the prime number 3: there's one 3 from the second term and one 3 from the third term. So total exponents for 3: 1 + 1 = 2.

For the prime number 5: there's one 5 from the first term and one 5 from the second term. So total exponents for 5: 1 + 1 = 2.

So putting that all together, the product is 2⁴ × 3² × 5². Let me check that again to make sure I didn't make a mistake.

Original factors:

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.

Yes, that seems correct. So the product is 2⁴ × 3² × 5².

Now, the problem asks for the positive square root of this product. So, sqrt(2⁴ × 3² × 5²).

To simplify the square root, I can take the square root of each factor separately. Remember that sqrt(a × b × c) = sqrt(a) × sqrt(b) × sqrt(c).

So, sqrt(2⁴) × sqrt(3²) × sqrt(5²).

Calculating each term:

sqrt(2⁴) = 2^(4/2) = 2² = 4.

sqrt(3²) = 3^(2/2) = 3^1 = 3.

sqrt(5²) = 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.

So, the square root of the product 10 × 15 × 24 is 60.

Wait, let me verify this by actually multiplying the original numbers and then taking the square root to make sure I didn't make a mistake.

First, compute 10 × 15. That's 150. Then, 150 × 24. Let me calculate that.

150 × 24. Well, 150 × 20 = 3000, and 150 × 4 = 600. So, 3000 + 600 = 3600.

So the product is 3600. Then, the square root of 3600 is... Well, 60 × 60 = 3600. So sqrt(3600) = 60.

Okay, that matches my previous result. So that seems correct.

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

Another way could be to pair numbers to make squares. For example, maybe 10 × 15 × 24 can be grouped in a way that makes perfect squares. Let me try that.

Looking at 10, 15, 24. Let me see if any of them can be paired.

Alternatively, maybe factor out common terms. Let me see.

Alternatively, notice that 10 × 15 = 150, and 150 × 24. Wait, that's what I did before.

Alternatively, maybe factor 10 × 15 × 24 as (10 × 24) × 15. Let me compute 10 × 24 first. 10 × 24 = 240. Then 240 × 15. Let me compute that.

240 × 10 = 2400, 240 × 5 = 1200, so 2400 + 1200 = 3600. Same result.

So sqrt(3600) = 60.

Alternatively, maybe there's a way to simplify before multiplying? Let me think.

Looking back at the prime factors: 2⁴ × 3² × 5². So sqrt(2⁴ × 3² × 5²) = 2² × 3 × 5 = 4 × 3 × 5 = 60.

Yes, same answer.

Alternatively, maybe I can take square roots step by step. Let me try that.

First, take sqrt(10 × 15 × 24). Maybe break it into sqrt(10 × 15) × sqrt(24).

But sqrt(10 × 15) is sqrt(150). Then sqrt(150) × sqrt(24).

But sqrt(150) is sqrt(25 × 6) = 5 × sqrt(6). Then sqrt(24) is sqrt(4 × 6) = 2 × sqrt(6).

So multiplying those together: 5 × sqrt(6) × 2 × sqrt(6) = 5 × 2 × sqrt(6) × sqrt(6) = 10 × 6 = 60.

Oh, that's another way to do it. So that also gives 60.

So regardless of the method, I get 60.

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 check once again.

Original product: 10 × 15 × 24.

Let me compute 10 × 15 first. 10 × 15 = 150. Then 150 × 24.

Breaking down 24 into 20 + 4. So 150 × 20 = 3000, 150 × 4 = 600. 3000 + 600 = 3600.

Square root of 3600. Well, 60 squared is 3600. 60 × 60 = 3600.

Yes, that's correct.

Alternatively, if I didn't remember that 60 squared is 3600, I could compute sqrt(3600).

Well, sqrt(3600) = sqrt(36 × 100) = sqrt(36) × sqrt(100) = 6 × 10 = 60.

Yes, that's another way.

So, all methods lead to 60.

Therefore, the answer 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 factoring each number into its prime components:

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

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:

- $2^1 \times 2^3 = 2^4$
- $3^1 \times 3^1 = 3^2$
- $5^1 \times 5^1 = 5^2$

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 $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can separate the square root:

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

Now simplify each term:

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

Now multiply the results:

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

---

### Step 3: Verification

To double-check, compute the product directly:

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

Then take the square root:

$$
\sqrt{3600} = 60
$$

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 can be written as 2 × 2 × 2 × 3, which is 2³ × 3.

So, putting it all together, the product 10 × 15 × 24 is equal to (2 × 5) × (3 × 5) × (2³ × 3). Let me write that out:

(2 × 5) × (3 × 5) × (2³ × 3) = 2 × 5 × 3 × 5 × 2³ × 3.

Now, let me combine like terms. Let me count the exponents for each prime factor.

For the prime number 2: there's one 2 from the first term and three 2s from the third term. So total exponents for 2: 1 + 3 = 4.

For the prime number 3: there's one 3 from the second term and one 3 from the third term. So total exponents for 3: 1 + 1 = 2.

For the prime number 5: there's one 5 from the first term and one 5 from the second term. So total exponents for 5: 1 + 1 = 2.

So putting that all together, the product is 2⁴ × 3² × 5². Let me check that again to make sure I didn't make a mistake.

Original factors:

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.

Yes, that seems correct. So the product is 2⁴ × 3² × 5².

Now, the problem asks for the positive square root of this product. So, sqrt(2⁴ × 3² × 5²).

To simplify the square root, I can take the square root of each factor separately. Remember that sqrt(a × b × c) = sqrt(a) × sqrt(b) × sqrt(c).

So, sqrt(2⁴) × sqrt(3²) × sqrt(5²).

Calculating each term:

sqrt(2⁴) = 2^(4/2) = 2² = 4.

sqrt(3²) = 3^(2/2) = 3^1 = 3.

sqrt(5²) = 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.

So, the square root of the product 10 × 15 × 24 is 60.

Wait, let me verify this by actually multiplying the original numbers and then taking the square root to make sure I didn't make a mistake.

First, compute 10 × 15. That's 150. Then, 150 × 24. Let me calculate that.

150 × 24. Well, 150 × 20 = 3000, and 150 × 4 = 600. So, 3000 + 600 = 3600.

So the product is 3600. Then, the square root of 3600 is... Well, 60 × 60 = 3600. So sqrt(3600) = 60.

Okay, that matches my previous result. So that seems correct.

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

Another way could be to pair numbers to make squares. For example, maybe 10 × 15 × 24 can be grouped in a way that makes perfect squares. Let me try that.

Looking at 10, 15, 24. Let me see if any of them can be paired.

Alternatively, maybe factor out common terms. Let me see.

Alternatively, notice that 10 × 15 = 150, and 150 × 24. Wait, that's what I did before.

Alternatively, maybe factor 10 × 15 × 24 as (10 × 24) × 15. Let me compute 10 × 24 first. 10 × 24 = 240. Then 240 × 15. Let me compute that.

240 × 10 = 2400, 240 × 5 = 1200, so 2400 + 1200 = 3600. Same result.

So sqrt(3600) = 60.

Alternatively, maybe there's a way to simplify before multiplying? Let me think.

Looking back at the prime factors: 2⁴ × 3² × 5². So sqrt(2⁴ × 3² × 5²) = 2² × 3 × 5 = 4 × 3 × 5 = 60.

Yes, same answer.

Alternatively, maybe I can take square roots step by step. Let me try that.

First, take sqrt(10 × 15 × 24). Maybe break it into sqrt(10 × 15) × sqrt(24).

But sqrt(10 × 15) is sqrt(150). Then sqrt(150) × sqrt(24).

But sqrt(150) is sqrt(25 × 6) = 5 × sqrt(6). Then sqrt(24) is sqrt(4 × 6) = 2 × sqrt(6).

So multiplying those together: 5 × sqrt(6) × 2 × sqrt(6) = 5 × 2 × sqrt(6) × sqrt(6) = 10 × 6 = 60.

Oh, that's another way to do it. So that also gives 60.

So regardless of the method, I get 60.

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 check once again.

Original product: 10 × 15 × 24.

Let me compute 10 × 15 first. 10 × 15 = 150. Then 150 × 24.

Breaking down 24 into 20 + 4. So 150 × 20 = 3000, 150 × 4 = 600. 3000 + 600 = 3600.

Square root of 3600. Well, 60 squared is 3600. 60 × 60 = 3600.

Yes, that's correct.

Alternatively, if I didn't remember that 60 squared is 3600, I could compute sqrt(3600).

Well, sqrt(3600) = sqrt(36 × 100) = sqrt(36) × sqrt(100) = 6 × 10 = 60.

Yes, that's another way.

So, all methods lead to 60.

Therefore, the answer 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 factoring each number into its prime components:

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

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:

- $2^1 \times 2^3 = 2^4$
- $3^1 \times 3^1 = 3^2$
- $5^1 \times 5^1 = 5^2$

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 $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$, we can separate the square root:

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

Now simplify each term:

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

Now multiply the results:

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

---

### Step 3: Verification

To double-check, compute the product directly:

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

Then take the square root:

$$
\sqrt{3600} = 60
$$

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, two-thirds of the total equals 834. Let me write that as an equation to make it clearer.

Let me denote the total number of students as "S". According to the problem, two-thirds of S is equal to 834. So, mathematically, that would be:

(2/3) * S = 834

Now, I need to solve for S. To isolate S, I should do the opposite operation of multiplying by 2/3, which is dividing by 2/3. 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.

S = 834 / (2/3) = 834 * (3/2)

Okay, so now I need to compute 834 multiplied by 3/2. Let me do that step by step. First, maybe I can simplify this multiplication. Multiplying by 3/2 is the same as multiplying by 3 and then dividing by 2, or dividing by 2 first and then multiplying by 3. Let me see which is easier.

Let me try dividing 834 by 2 first. 834 divided by 2 is 417. Then, multiplying that by 3. So, 417 * 3. Let me calculate that. 400*3 is 1200, and 17*3 is 51, so adding those together gives 1200 + 51 = 1251. So, S = 1251?

Wait, let me check that again. If I take 834 divided by 2, that's 417. Then 417 times 3. Let me do 400*3 = 1200, 17*3=51, so 1200+51=1251. Yeah, that seems right.

Alternatively, I could have done 834 * 3 first, which is 2502, and then divided by 2, which is also 1251. Either way, same result. So that seems consistent.

Let me verify if this answer makes sense. If the total number of students is 1251, then two-thirds of that should be 834. Let me calculate 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 12, bring down 5. 3 goes into 5 once, 1*3=3, subtract 3, remainder 2. Bring down 1, making 21. 3 goes into 21 seven times. So, 1251 divided by 3 is 417. Then, multiplying by 2 gives 417*2=834. Perfect, that matches the given number of students taking music. So that checks out.

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

Wait, but let me just make sure I didn't make any calculation errors. Let me recheck the multiplication steps.

Original equation: (2/3) * S = 834. Solving for S: S = 834 * (3/2).

Calculating 834 * 3: 800*3=2400, 34*3=102, so 2400 + 102 = 2502. Then divide by 2: 2502 / 2 = 1251. Yep, same result.

Alternatively, breaking down 834 into 800 + 34. So, 800*(3/2) + 34*(3/2). 800*(3/2) is 400*3 = 1200. 34*(3/2) is 17*3 = 51. 1200 + 51 = 1251. Yep, same answer.

So, all methods lead to 1251. And the check by plugging back in works. So, I think that's correct.

Is there another way to think about this? Maybe using proportions. If two-thirds correspond to 834 students, then one-third would correspond to half of that, which is 417. Then, the total number of students would be three times that, which is 417*3=1251. Yep, same answer.

So, all different approaches confirm that the total number of students is 1251. Therefore, I can be confident that this 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 number is **834**. Our goal is to find the **total number of students**, which we'll denote as $ S $.

---

### Step 1: Set Up the Equation

We are told that:

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

This equation expresses that two-thirds of the total number of students equals 834.

---

### Step 2: Solve for $ S $

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

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

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 check by plugging $ S = 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, two-thirds of the total equals 834. Let me write that as an equation to make it clearer.

Let me denote the total number of students as "S". According to the problem, two-thirds of S is equal to 834. So, mathematically, that would be:

(2/3) * S = 834

Now, I need to solve for S. To isolate S, I should do the opposite operation of multiplying by 2/3, which is dividing by 2/3. 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.

S = 834 / (2/3) = 834 * (3/2)

Okay, so now I need to compute 834 multiplied by 3/2. Let me do that step by step. First, maybe I can simplify this multiplication. Multiplying by 3/2 is the same as multiplying by 3 and then dividing by 2, or dividing by 2 first and then multiplying by 3. Let me see which is easier.

Let me try dividing 834 by 2 first. 834 divided by 2 is 417. Then, multiplying that by 3. So, 417 * 3. Let me calculate that. 400*3 is 1200, and 17*3 is 51, so adding those together gives 1200 + 51 = 1251. So, S = 1251?

Wait, let me check that again. If I take 834 divided by 2, that's 417. Then 417 times 3. Let me do 400*3 = 1200, 17*3=51, so 1200+51=1251. Yeah, that seems right.

Alternatively, I could have done 834 * 3 first, which is 2502, and then divided by 2, which is also 1251. Either way, same result. So that seems consistent.

Let me verify if this answer makes sense. If the total number of students is 1251, then two-thirds of that should be 834. Let me calculate 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 12, bring down 5. 3 goes into 5 once, 1*3=3, subtract 3, remainder 2. Bring down 1, making 21. 3 goes into 21 seven times. So, 1251 divided by 3 is 417. Then, multiplying by 2 gives 417*2=834. Perfect, that matches the given number of students taking music. So that checks out.

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

Wait, but let me just make sure I didn't make any calculation errors. Let me recheck the multiplication steps.

Original equation: (2/3) * S = 834. Solving for S: S = 834 * (3/2).

Calculating 834 * 3: 800*3=2400, 34*3=102, so 2400 + 102 = 2502. Then divide by 2: 2502 / 2 = 1251. Yep, same result.

Alternatively, breaking down 834 into 800 + 34. So, 800*(3/2) + 34*(3/2). 800*(3/2) is 400*3 = 1200. 34*(3/2) is 17*3 = 51. 1200 + 51 = 1251. Yep, same answer.

So, all methods lead to 1251. And the check by plugging back in works. So, I think that's correct.

Is there another way to think about this? Maybe using proportions. If two-thirds correspond to 834 students, then one-third would correspond to half of that, which is 417. Then, the total number of students would be three times that, which is 417*3=1251. Yep, same answer.

So, all different approaches confirm that the total number of students is 1251. Therefore, I can be confident that this 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 number is **834**. Our goal is to find the **total number of students**, which we'll denote as $ S $.

---

### Step 1: Set Up the Equation

We are told that:

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

This equation expresses that two-thirds of the total number of students equals 834.

---

### Step 2: Solve for $ S $

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

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

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 check by plugging $ S = 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 + 2b = 12. 3*4 is indeed 12, so 12 + 2b = 12. Subtract 12 from both sides: 2b = 0. Divide by 2: b = 0. That seems straightforward.

But let me verify by plugging the values back into the original equation. If a = 4 and b = 0, then the point is (4, 0). Let's check if this satisfies 3x + 2y = 12.

Calculating 3*4 + 2*0 = 12 + 0 = 12. Which is equal to the right side of the equation, 12. So yes, that works.

Hmm, seems like everything checks out. Maybe I was overcomplicating it, but it's good to verify. So the value of b when a = 4 is 0.

Alternatively, maybe I can think about the equation in slope-intercept form to see if that gives the same result. Let me try converting 3x + 2y = 12 into y = mx + c form.

Starting with 3x + 2y = 12. Subtract 3x from both sides: 2y = -3x + 12. Then divide both sides by 2: y = (-3/2)x + 6.

So the equation in slope-intercept form is y = -3/2 x + 6. Now, if x is 4, then y would be:

y = (-3/2)*(4) + 6. Let me compute that.

First, (-3/2)*4 is (-3*4)/2 = -12/2 = -6. Then adding 6 gives -6 + 6 = 0. So y = 0.

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, so I'm confident that b = 0 is correct.

Another way to think about it: maybe graphing? If I imagine the line 3x + 2y = 12, it's a straight line. When x is 0, y is 6, and when y is 0, x is 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's another confirmation. So regardless of the method I use—substitution, converting to slope-intercept form, or even thinking about intercepts—the answer is the same.

I don't see any mistakes in my reasoning. All steps are logical and consistent. Therefore, I can conclude that the value of b is indeed 0.

**Final Answer**
The value of $ b $ when $ a = 4 $ 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: Substitute the known value of $ a $ into the equation

Since the point $ (a, b) $ lies on the line, its coordinates must satisfy the equation. That is:

$$
3x + 2y = 12
$$

Substitute $ x = a = 4 $ and $ y = b $:

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

---

### Step 2: Simplify the equation

$$
12 + 2b = 12
$$

Subtract 12 from both sides:

$$
2b = 0
$$

---

### Step 3: Solve for $ b $

$$
b = \frac{0}{2} = 0
$$

---

### Step 4: 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 + 2b = 12. 3*4 is indeed 12, so 12 + 2b = 12. Subtract 12 from both sides: 2b = 0. Divide by 2: b = 0. That seems straightforward.

But let me verify by plugging the values back into the original equation. If a = 4 and b = 0, then the point is (4, 0). Let's check if this satisfies 3x + 2y = 12.

Calculating 3*4 + 2*0 = 12 + 0 = 12. Which is equal to the right side of the equation, 12. So yes, that works.

Hmm, seems like everything checks out. Maybe I was overcomplicating it, but it's good to verify. So the value of b when a = 4 is 0.

Alternatively, maybe I can think about the equation in slope-intercept form to see if that gives the same result. Let me try converting 3x + 2y = 12 into y = mx + c form.

Starting with 3x + 2y = 12. Subtract 3x from both sides: 2y = -3x + 12. Then divide both sides by 2: y = (-3/2)x + 6.

So the equation in slope-intercept form is y = -3/2 x + 6. Now, if x is 4, then y would be:

y = (-3/2)*(4) + 6. Let me compute that.

First, (-3/2)*4 is (-3*4)/2 = -12/2 = -6. Then adding 6 gives -6 + 6 = 0. So y = 0.

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, so I'm confident that b = 0 is correct.

Another way to think about it: maybe graphing? If I imagine the line 3x + 2y = 12, it's a straight line. When x is 0, y is 6, and when y is 0, x is 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's another confirmation. So regardless of the method I use—substitution, converting to slope-intercept form, or even thinking about intercepts—the answer is the same.

I don't see any mistakes in my reasoning. All steps are logical and consistent. Therefore, I can conclude that the value of b is indeed 0.

**Final Answer**
The value of $ b $ when $ a = 4 $ 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: Substitute the known value of $ a $ into the equation

Since the point $ (a, b) $ lies on the line, its coordinates must satisfy the equation. That is:

$$
3x + 2y = 12
$$

Substitute $ x = a = 4 $ and $ y = b $:

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

---

### Step 2: Simplify the equation

$$
12 + 2b = 12
$$

Subtract 12 from both sides:

$$
2b = 0
$$

---

### Step 3: Solve for $ b $

$$
b = \frac{0}{2} = 0
$$

---

### Step 4: 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... Let me think. 36 minus 20 would be 16, but since it's 19, which is 1 less than 20, I add 1 back. So, 16 + 1 = 17. Therefore, x equals 17.

Wait, let me check my steps again to make sure I didn't make a mistake.

Starting with the mean formula: 9 = (7 + 2 + x + 10)/4. Adding 7 + 2 + 10 gives 19, so 9 = (19 + x)/4. Multiply both sides by 4: 36 = 19 + x. Subtract 19: 36 - 19 = 17. So x = 17. That seems right.

But maybe I should verify by plugging the value back into the original equation to ensure it works. Let me do that.

If x is 17, then the numbers are 7, 2, 17, and 10. Adding them up: 7 + 2 is 9, 9 + 17 is 26, 26 + 10 is 36. Then divide by 4: 36 / 4 = 9. Which matches the given mean. So that checks out.

Hmm, seems like everything is consistent. So x must be 17.

Wait, but let me think again if there's another way to approach this problem. Maybe by using algebraic manipulation differently? Let me try.

Starting again, the mean is 9. So total sum should be mean multiplied by number of terms. That is, total sum = mean * number of terms. So total sum = 9 * 4 = 36. Therefore, the sum of 7 + 2 + x + 10 must equal 36. Which is the same equation as before. So 7 + 2 + 10 is 19, so 19 + x = 36. Therefore, x = 36 - 19 = 17. Yep, same result.

Another way to check: maybe think about what number would make the average 9. The numbers are 7, 2, 10, and x. Let me think about the current numbers without x. 7, 2, 10. Their sum is 19. If I want the average of four numbers to be 9, the total needs to be 36. So the missing number is 36 - 19 = 17.

Alternatively, maybe think about how much each number contributes. The average is 9, so each number should be around 9. The numbers given are 7, 2, 10. 7 is 2 less than 9, 2 is 7 less than 9, 10 is 1 more than 9. So the total deficit and surplus should balance out.

So, 7 is -2, 2 is -7, 10 is +1. So total deviation from 9 is (-2) + (-7) + 1 = -8. Therefore, the fourth number x must make up for this -8 deviation. Since the average is 9, x should be 9 + 8 = 17. Because if the total deviation is -8, then x needs to be 8 more than 9 to balance it out.

That's another way to see it. So that also gives x = 17.

So all methods lead to x being 17. Therefore, I can be confident that the answer is 17.

But just to make sure, let me consider if there's any possibility of a mistake in arithmetic. For example, when adding 7 + 2 + 10. 7 + 2 is 9, 9 + 10 is 19. Correct. Then 36 - 19. 36 - 10 is 26, 26 - 9 is 17. Correct.

Alternatively, 19 + x = 36. So x = 36 - 19. 36 - 19: 19 + 17 = 36. Yes, because 19 + 10 = 29, 29 + 7 = 36. So 17 is correct.

Another check: if x was something else, say 18, then total sum would be 19 + 18 = 37, average would be 37/4 = 9.25, which is higher than 9. If x was 16, total sum is 19 + 16 = 35, average is 35/4 = 8.75, which is lower. So 17 is exactly in between, giving 9.

Therefore, all checks confirm that x is 17.

I think that's thorough enough. I don't see any mistakes in my reasoning. So the answer should be 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: Use the formula for arithmetic mean

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

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

So we can write:

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

---

### Step 2: Simplify the equation

First, add the known numbers:

$$
7 + 2 + 10 = 19
$$

So the equation becomes:

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

---

### Step 3: Solve for $ x $

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

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

$$
36 = 19 + x
$$

Now subtract 19 from both sides:

$$
x = 36 - 19 = 17
$$

---

### Step 4: Verify the solution

Plug $ x = 17 $ back into the original set of numbers: 7, 2, 17, 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... Let me think. 36 minus 20 would be 16, but since it's 19, which is 1 less than 20, I add 1 back. So, 16 + 1 = 17. Therefore, x equals 17.

Wait, let me check my steps again to make sure I didn't make a mistake.

Starting with the mean formula: 9 = (7 + 2 + x + 10)/4. Adding 7 + 2 + 10 gives 19, so 9 = (19 + x)/4. Multiply both sides by 4: 36 = 19 + x. Subtract 19: 36 - 19 = 17. So x = 17. That seems right.

But maybe I should verify by plugging the value back into the original equation to ensure it works. Let me do that.

If x is 17, then the numbers are 7, 2, 17, and 10. Adding them up: 7 + 2 is 9, 9 + 17 is 26, 26 + 10 is 36. Then divide by 4: 36 / 4 = 9. Which matches the given mean. So that checks out.

Hmm, seems like everything is consistent. So x must be 17.

Wait, but let me think again if there's another way to approach this problem. Maybe by using algebraic manipulation differently? Let me try.

Starting again, the mean is 9. So total sum should be mean multiplied by number of terms. That is, total sum = mean * number of terms. So total sum = 9 * 4 = 36. Therefore, the sum of 7 + 2 + x + 10 must equal 36. Which is the same equation as before. So 7 + 2 + 10 is 19, so 19 + x = 36. Therefore, x = 36 - 19 = 17. Yep, same result.

Another way to check: maybe think about what number would make the average 9. The numbers are 7, 2, 10, and x. Let me think about the current numbers without x. 7, 2, 10. Their sum is 19. If I want the average of four numbers to be 9, the total needs to be 36. So the missing number is 36 - 19 = 17.

Alternatively, maybe think about how much each number contributes. The average is 9, so each number should be around 9. The numbers given are 7, 2, 10. 7 is 2 less than 9, 2 is 7 less than 9, 10 is 1 more than 9. So the total deficit and surplus should balance out.

So, 7 is -2, 2 is -7, 10 is +1. So total deviation from 9 is (-2) + (-7) + 1 = -8. Therefore, the fourth number x must make up for this -8 deviation. Since the average is 9, x should be 9 + 8 = 17. Because if the total deviation is -8, then x needs to be 8 more than 9 to balance it out.

That's another way to see it. So that also gives x = 17.

So all methods lead to x being 17. Therefore, I can be confident that the answer is 17.

But just to make sure, let me consider if there's any possibility of a mistake in arithmetic. For example, when adding 7 + 2 + 10. 7 + 2 is 9, 9 + 10 is 19. Correct. Then 36 - 19. 36 - 10 is 26, 26 - 9 is 17. Correct.

Alternatively, 19 + x = 36. So x = 36 - 19. 36 - 19: 19 + 17 = 36. Yes, because 19 + 10 = 29, 29 + 7 = 36. So 17 is correct.

Another check: if x was something else, say 18, then total sum would be 19 + 18 = 37, average would be 37/4 = 9.25, which is higher than 9. If x was 16, total sum is 19 + 16 = 35, average is 35/4 = 8.75, which is lower. So 17 is exactly in between, giving 9.

Therefore, all checks confirm that x is 17.

I think that's thorough enough. I don't see any mistakes in my reasoning. So the answer should be 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: Use the formula for arithmetic mean

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

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

So we can write:

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

---

### Step 2: Simplify the equation

First, add the known numbers:

$$
7 + 2 + 10 = 19
$$

So the equation becomes:

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

---

### Step 3: Solve for $ x $

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

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

$$
36 = 19 + x
$$

Now subtract 19 from both sides:

$$
x = 36 - 19 = 17
$$

---

### Step 4: Verify the solution

Plug $ x = 17 $ back into the original set of numbers: 7, 2, 17, 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, $(m,b) = \boxed{\left( \frac{3}{2}, -13 \right)}.$


Explanation:
Okay, so Mikka wants to order a pizza with two different toppings, and he has 8 different toppings to choose from. The question is asking how many different pizzas he could order. Hmm, let me think about this step by step.

First, I need to figure out if this is a combination or permutation problem. Since the order of the toppings doesn't matter on a pizza, right? Like, having pepperoni and mushrooms is the same as mushrooms and pepperoni. So, the order isn't important here. That means it's a combination problem, not a permutation.

Combinations are used when the order doesn't matter, and permutations when it does. So, combinations are calculated using the formula C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items to choose. In this case, n is 8 toppings, and k is 2 toppings.

Let me write that down. So, C(8, 2) = 8! / (2! * (8 - 2)!) = 8! / (2! * 6!).

But wait, maybe I can simplify this without calculating the entire factorials. Since 8! is 8 × 7 × 6!, right? So, if I write that out, it becomes (8 × 7 × 6!) / (2! × 6!). Then the 6! cancels out from numerator and denominator. So, that leaves me with (8 × 7) / 2!.

And 2! is 2 × 1 = 2. So, that simplifies to (8 × 7) / 2. Let me compute that. 8 times 7 is 56. Then 56 divided by 2 is 28. So, does that mean there are 28 different ways to choose two toppings?

Wait, let me check if I did that right. Another way to think about combinations is that for each topping, you can pair it with another topping. So, for the first topping, there are 8 choices, and then for the second topping, there are 7 choices left. But since order doesn't matter, each pair is counted twice in this method. So, the total number would be (8 × 7) / 2, which is again 56 / 2 = 28. Yep, same answer.

So, that seems to confirm it. So, the answer is 28 different pizzas.

But let me think again if there's another way to approach this. Maybe using the combination formula directly. Let me recall that C(n, k) is the number of ways to choose k elements from a set of n elements without regard to order. So, with 8 toppings, choosing 2, it's definitely C(8, 2).

Alternatively, if I list out the toppings as A, B, C, D, E, F, G, H. Then, how many pairs can I make? Let's see. Starting with A, I can pair it with B, C, D, E, F, G, H. That's 7 pairs. Then with B, I already paired it with A, so the new pairs are C, D, E, F, G, H. That's 6. Then with C, it's D, E, F, G, H. 5. Then D: E, F, G, H. 4. E: F, G, H. 3. F: G, H. 2. G: H. 1. And H has already been paired with everyone. So, adding those up: 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28.

Yes, that's another way to see it. So, that also gives 28. So, that's consistent.

Therefore, I think the answer is 28.

But just to make sure, let me think if there's any possibility that the problem is considering something else. For example, maybe the pizza can have two different toppings, but maybe some toppings are considered the same? But the problem says "two different toppings" and "8 different toppings to choose from." So, each topping is unique, and he wants to choose two different ones. So, no repetition.

So, no, there's no issue with repetition here. So, combinations without repetition.

Alternatively, if the problem had said that he can choose the same topping twice, then it would be a different story. But since it says two different toppings, it's combinations without repetition.

Therefore, the answer is 28.

Another check: the formula for combinations is n(n - 1)/2 when k = 2. Which is exactly what we did. So, 8*7/2 = 28.

So, yeah, that's straightforward.

I think that's solid. So, Mikka can order 28 different pizzas with two different toppings.

**Final Answer**
The number of different pizzas Mikka could order is \boxed{28}.

To determine how many different pizzas Mikka could order, we need to analyze the situation carefully.

---

### **Understanding the Problem**

Mikka wants to order a pizza with **two different toppings**, and he has **8 different toppings** to choose from. The key point here is that the **order of toppings does not matter** on a pizza. That is, a pizza with toppings A and B is the same as one with B and A. Therefore, this is a **combinations** problem, not a permutations problem.

---

### **Choosing Two Toppings from Eight**

We are selecting **2 toppings** from a total of **8 toppings**, and the order does not matter. The number of such combinations is given by the **binomial coefficient**:

$$
\binom{8}{2} = \frac{8!}{2!(8 - 2)!}
$$

Simplifying this:

$$
\binom{8}{2} = \frac{8 \times 7 \times 6!}{2 \times 1 \times 6!} = \frac{8 \times 7}{2} = \frac{56}{2} = 28
$$

---

### **Alternative Verification**

We can also verify this by thinking of it as pairing each of the 8 toppings with the others. For the first topping, there are 7 possible pairings. For the second, 6, and so on, down to 1. This gives:

$$
7 + 6 + 5 + 4 + 3 + 2 + 1 = 28
$$

This confirms the result.

---

### **Conclusion**

Since the order of toppings doesn't matter and we are choosing 2 toppings from 8, the number of different pizzas Mikka could order is:

$$
\boxed{28}
$$

[Assistant]
Explanation:
Okay, so Mikka wants to order a pizza with two different toppings, and he has 8 different toppings to choose from. The question is asking how many different pizzas he could order. Hmm, let me think about this step by step.

First, I need to figure out if this is a combination or permutation problem. Since the order of the toppings doesn't matter on a pizza, right? Like, having pepperoni and mushrooms is the same as mushrooms and pepperoni. So, the order isn't important here. That means it's a combination problem, not a permutation.

Combinations are used when the order doesn't matter, and permutations when it does. So, combinations are calculated using the formula C(n, k) = n! / (k! * (n - k)!), where n is the total number of items, and k is the number of items to choose. In this case, n is 8 toppings, and k is 2 toppings.

Let me write that down. So, C(8, 2) = 8! / (2! * (8 - 2)!) = 8! / (2! * 6!).

But wait, maybe I can simplify this without calculating the entire factorials. Since 8! is 8 × 7 × 6!, right? So, if I write that out, it becomes (8 × 7 × 6!) / (2! × 6!). Then the 6! cancels out from numerator and denominator. So, that leaves me with (8 × 7) / 2!.

And 2! is 2 × 1 = 2. So, that simplifies to (8 × 7) / 2. Let me compute that. 8 times 7 is 56. Then 56 divided by 2 is 28. So, does that mean there are 28 different ways to choose two toppings?

Wait, let me check if I did that right. Another way to think about combinations is that for each topping, you can pair it with another topping. So, for the first topping, there are 8 choices, and then for the second topping, there are 7 choices left. But since order doesn't matter, each pair is counted twice in this method. So, the total number would be (8 × 7) / 2, which is again 56 / 2 = 28. Yep, same answer.

So, that seems to confirm it. So, the answer is 28 different pizzas.

But let me think again if there's another way to approach this. Maybe using the combination formula directly. Let me recall that C(n, k) is the number of ways to choose k elements from a set of n elements without regard to order. So, with 8 toppings, choosing 2, it's definitely C(8, 2).

Alternatively, if I list out the toppings as A, B, C, D, E, F, G, H. Then, how many pairs can I make? Let's see. Starting with A, I can pair it with B, C, D, E, F, G, H. That's 7 pairs. Then with B, I already paired it with A, so the new pairs are C, D, E, F, G, H. That's 6. Then with C, it's D, E, F, G, H. 5. Then D: E, F, G, H. 4. E: F, G, H. 3. F: G, H. 2. G: H. 1. And H has already been paired with everyone. So, adding those up: 7 + 6 + 5 + 4 + 3 + 2 + 1 = 28.

Yes, that's another way to see it. So, that also gives 28. So, that's consistent.

Therefore, I think the answer is 28.

But just to make sure, let me think if there's any possibility that the problem is considering something else. For example, maybe the pizza can have two different toppings, but maybe some toppings are considered the same? But the problem says "two different toppings" and "8 different toppings to choose from." So, each topping is unique, and he wants to choose two different ones. So, no repetition.

So, no, there's no issue with repetition here. So, combinations without repetition.

Alternatively, if the problem had said that he can choose the same topping twice, then it would be a different story. But since it says two different toppings, it's combinations without repetition.

Therefore, the answer is 28.

Another check: the formula for combinations is n(n - 1)/2 when k = 2. Which is exactly what we did. So, 8*7/2 = 28.

So, yeah, that's straightforward.

I think that's solid. So, Mikka can order 28 different pizzas with two different toppings.

**Final Answer**
The number of different pizzas Mikka could order is \boxed{28}.

To determine how many different pizzas Mikka could order, we need to analyze the situation carefully.

---

### **Understanding the Problem**

Mikka wants to order a pizza with **two different toppings**, and he has **8 different toppings** to choose from. The key point here is that the **order of toppings does not matter** on a pizza. That is, a pizza with toppings A and B is the same as one with B and A. Therefore, this is a **combinations** problem, not a permutations problem.

---

### **Choosing Two Toppings from Eight**

We are selecting **2 toppings** from a total of **8 toppings**, and the order does not matter. The number of such combinations is given by the **binomial coefficient**:

$$
\binom{8}{2} = \frac{8!}{2!(8 - 2)!}
$$

Simplifying this:

$$
\binom{8}{2} = \frac{8 \times 7 \times 6!}{2 \times 1 \times 6!} = \frac{8 \times 7}{2} = \frac{56}{2} = 28
$$

---

### **Alternative Verification**

We can also verify this by thinking of it as pairing each of the 8 toppings with the others. For the first topping, there are 7 possible pairings. For the second, 6, and so on, down to 1. This gives:

$$
7 + 6 + 5 + 4 + 3 + 2 + 1 = 28
$$

This confirms the result.

---

### **Conclusion**

Since the order of toppings doesn't matter and we are choosing 2 toppings from 8, the number of different pizzas Mikka could order is:

$$
\boxed{28}
$$

Solution: There are 8 choices of toppings, and we need to choose 2 distinct toppings. This is represented by the number of 2-element subsets of an 8 -element set. We use the binomial coefficient ${8 \choose 2} = \boxed{28}$ to compute this.