The senior classes at High School A and High School B planned separate trips to the county fair.

The senior class at High School A rented and filled 1 van and 7 buses with 197 students.

 High School B rented and filled 12 vans and 1 bus with 123 students.  

Each van and each bus carried the same number of students.

 Find the number of students in each van and in each bus.

  Ans.

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

B.    Number of students in VAN =  18   and  Number of students in a BUS=  20

C.    Number of students in VAN =  15   and  Number of students in a BUS=  27

D.    Number of students in VAN =  8   and  Number of students in a BUS=  20

   

Group of answer choices

C.    Number of students in VAN =  15   and  Number of students in a BUS=  27

B.    Number of students in VAN =  18   and  Number of students in a BUS=  20

D.    Number of students in VAN =  8   and  Number of students in a BUS=  20

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

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**

---

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.

The senior classes at High School A and High School B planned separate trips to the county fair. The senior class at High School A rented and filled 1 van and 7 buses with 197 students. High School B rented and filled 12 vans and 1 bus with 123 students. Each van and each bus carried the same number of students. Find the number of students in each van and in each bus.

Learn how to solve a system of linear equations to determine the number of students in each van and each bus. This problem involves two equations based on student distribution between vans and buses from two high schools, and it is solved using substitution. Ideal for students in Grades 9-10.

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