The senior classes at High School A and High School B planned separate trips to the indoor climbing gym. The senior class at High School A rented and filled 43 vans and 29 buses with 1780 students. High School B rented and filled 38 vans and 21 buses with 1337 students. Each van and each bus carried the same number of students. Find the number of students in each van and in each bus.



To solve this problem, we will set up a system of linear equations based on the information given.

### Define Variables:
- Let \( x \) be the number of students per van.
- Let \( y \) be the number of students per bus.

### Step 1: Set Up Equations

From the information provided:
1. **High School A**: 43 vans and 29 buses carry 1780 students.
   \[
   43x + 29y = 1780
   \]

2. **High School B**: 38 vans and 21 buses carry 1337 students.
   \[
   38x + 21y = 1337
   \]

### Step 2: Solve the System of Equations

We can solve these two equations using the elimination or substitution method. Here, we'll use the elimination method to eliminate one of the variables.

Let's multiply the first equation by 21 and the second equation by 29 to align the coefficients of \( y \):

\[
21(43x + 29y) = 21(1780)
\]

\[
29(38x + 21y) = 29(1337)
\]

This simplifies to:

\[
903x + 609y = 37380
\]

\[
1102x + 609y = 38773
\]

Now subtract the first equation from the second:

\[
(1102x + 609y) - (903x + 609y) = 38773 - 37380
\]

\[
199x = 1393
\]

\[
x = \frac{1393}{199} = 7
\]

So, there are 7 students per van.

### Step 3: Substitute to Find \( y \)

Substitute \( x = 7 \) back into one of the original equations. Let's use the first one:

\[
43(7) + 29y = 1780
\]

\[
301 + 29y = 1780
\]

\[
29y = 1479
\]

\[
y = \frac{1479}{29} = 51
\]

So, there are 51 students per bus.

### Conclusion:

- **Number of students per van (x):** 7  
- **Number of students per bus (y):** 51  

### Verification:

Substitute \( x = 7 \) and \( y = 51 \) into the second equation to check:

\[
38(7) + 21(51) = 1337
\]

\[
266 + 1071 = 1337
\]

The equation holds true, confirming our solution is correct.

### Would you like a detailed explanation of each step, or do you have any other questions?

#### Related Questions:
1. What other methods could we use to solve the system of equations?
2. How would the solution change if there were more types of vehicles?
3. How does changing the number of students in a van affect the total?
4. Can we graphically represent this problem?
5. How do we interpret the results if the number of students must be different?

#### Tip:
When solving linear equations, always check your solutions by substituting them back into the original equations.

The senior classes at High School A and High School B planned separate trips to the indoor climbing gym. The senior class at High School A rented and filled 43 vans and 29 buses with 1780 students. High School B rented and filled 38 vans and 21 buses with 1337 students. Each van and each bus carried the same number of students. Find the number of students in each van and in each bus.

Discover how to solve a real-life problem involving systems of linear equations. High School A and B filled 43 vans and 29 buses with 1780 students and 38 vans and 21 buses with 1337 students, respectively. Learn to find the number of students per van and bus with a detailed, step-by-step solution. Suitable for students in Grades 9-11.

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