New York City is a popular field trip destination. This year the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 5 vans and 3 buses with 240 students. High School B rented and filled 10 vans and 8 buses with 580 students. Every van had the same number of students in it as did the buses. What is the solution to the system of equations? Find the number of students in each van and in each bus.
Question
New York City is a popular field trip destination. This year the senior class at High School A and the senior class at High School B both planned trips there. The senior class at High School A rented and filled 5 vans and 3 buses with 240 students. High School B rented and filled 10 vans and 8 buses with 580 students. Every van had the same number of students in it as did the buses. What is the solution to the system of equations? Find the number of students in each van and in each bus.
Solution
Let's denote the number of students in each van as "v" and the number of students in each bus as "b".
From the problem, we can create two equations based on the information given:
For High School A: 5v + 3b = 240
For High School B: 10v + 8b = 580
This is a system of linear equations. To solve it, we can use either substitution or elimination method. Here, we will use the elimination method.
First, we can multiply the first equation by 2 to make the coefficients of 'b' the same in both equations:
10v + 6b = 480
Now we have: 10v + 6b = 480 10v + 8b = 580
Subtract the first equation from the second:
2b = 100
Solve for 'b': b = 100 / 2 = 50
Substitute 'b' = 50 into the first equation:
5v + 3*50 = 240 5v + 150 = 240 5v = 240 - 150 5v = 90 v = 90 / 5 v = 18
So, each van and each bus had 18 students and 50 students respectively.
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