Question

1. work required for full credit! (a.ei.2, 10 points)

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 3 vans and 12 buses with 474 students.
high school b rented and filled 6 vans and 14 buses with 578 students. each van and each bus
carried the same number of students. find the number of students in each van and in each bus.
) van: 10, bus: 37 b) van: 16, bus: 45 c) van: 6, bus: 43 d) van: 13, bus:

Explanation:

Step1: Define Variables

Let \( v \) be the number of students in a van and \( b \) be the number of students in a bus.

Step2: Set Up Equations

From High School A: \( 3v + 12b = 474 \) (simplify: \( v + 4b = 158 \) or \( v = 158 - 4b \))
From High School B: \( 6v + 14b = 578 \)

Step3: Substitute \( v \)

Substitute \( v = 158 - 4b \) into the second equation:
\( 6(158 - 4b) + 14b = 578 \)
\( 948 - 24b + 14b = 578 \)
\( 948 - 10b = 578 \)
\( -10b = 578 - 948 \)
\( -10b = -370 \)
\( b = 37 \)

Step4: Find \( v \)

Substitute \( b = 37 \) into \( v = 158 - 4b \):
\( v = 158 - 4(37) = 158 - 148 = 10 \)

Step5: Check Options

The values \( v = 10 \), \( b = 37 \) match option A.

Answer:

A. Van: 10, Bus: 37