Let's solve the problem step-by-step using a system of equations.

Step 1: Define Variables

Let:

• $v$ = number of students in each van.
• $b$ = number of students in each bus.

Step 2: Set Up the Equations

1. For High School A: 1 van and 7 buses were filled with 197 students. $v + 7b = 197$

2. For High School B: 12 vans and 1 bus were filled with 123 students. $12v + b = 123$

Step 3: Solve the System of Equations

We have the following system of equations:

1. $v + 7b = 197$
2. $12v + b = 123$

Let's solve equation (1) for $v$:

$v = 197 - 7b$

Substitute $v = 197 - 7b$ into equation (2):

$12(197 - 7b) + b = 123$

Simplify:

$2364 - 84b + b = 123$ $2364 - 83b = 123$ $-83b = 123 - 2364$ $-83b = -2241$ $b = \frac{-2241}{-83} = 27$

So, the number of students in each bus is 27.

Now, substitute $b = 27$ back into $v = 197 - 7b$:

$v = 197 - 7(27)$ $v = 197 - 189$ $v = 8$

So, the number of students in each van is 8.

Final Answer

• Number of students in each van = 8
• Number of students in each bus = 27

This corresponds to option A:

A. Number of students in VAN = 8 and Number of students in a BUS = 27

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Would you like to see more details or have any questions? Here are some related questions for further practice:

1. How do you verify the solution for a system of linear equations?
2. What is the method of substitution in solving linear systems?
3. How can we use matrix methods (like Gaussian elimination) to solve this problem?
4. What is the graphical interpretation of a system of equations like this?
5. Can we solve the problem using elimination instead of substitution?

Tip: Always verify your solution by substituting the values back into the original equations to ensure they satisfy both conditions.