So, to find the number of students each type of vehicle will carry, we are given the system of equations:
[tex]\[
\begin{cases}
4v + 8b = 488 \\
v + 9b = 500
\end{cases}
\][/tex]
We need to determine the values of [tex]\( v \)[/tex] and [tex]\( b \)[/tex]. Here is the step-by-step solution:
1. Solve the second equation for [tex]\( v \)[/tex]:
[tex]\[
v + 9b = 500
\][/tex]
[tex]\[
v = 500 - 9b
\][/tex]
2. Substitute [tex]\( v = 500 - 9b \)[/tex] into the first equation:
[tex]\[
4v + 8b = 488
\][/tex]
[tex]\[
4(500 - 9b) + 8b = 488
\][/tex]
Simplify the expression:
[tex]\[
2000 - 36b + 8b = 488
\][/tex]
Combine like terms:
[tex]\[
2000 - 28b = 488
\][/tex]
3. Isolate the variable [tex]\( b \)[/tex]:
[tex]\[
2000 - 488 = 28b
\][/tex]
[tex]\[
1512 = 28b
\][/tex]
Divide both sides by 28:
[tex]\[
b = \frac{1512}{28}
\][/tex]
[tex]\[
b = 54
\][/tex]
4. Substitute [tex]\( b = 54 \)[/tex] back into the equation [tex]\( v = 500 - 9b \)[/tex]:
[tex]\[
v = 500 - 9(54)
\][/tex]
[tex]\[
v = 500 - 486
\][/tex]
[tex]\[
v = 14
\][/tex]
Therefore, the number of students each type of vehicle will carry is:
- [tex]\( v = 14 \)[/tex] students per van
- [tex]\( b = 54 \)[/tex] students per bus
So, the final answer is:
- Each van will carry 14 students.
- Each bus will carry 54 students.
