Systems of Equations Word Problem

by Sam
(Usaexeter, RI)

All 231 students in the french class went on a field trip. Some students rode in vans which hold 7 students each and some students rode in buses which hold 25 students each. How many of each type of vehicle did they use if there were 15 vehicles total?

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Mar 04, 2010
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by: Karin

In this problem, you need to ask yourself: What do I need to find?

You need to find how many buses and how many vans. There are two things that you don't know. Let's assign our variables:

Let x = number of vans
Let y = number of buses

Typically when there are two variables, you will need to write a system of equations. You will need to write two different equations and then solve by using substitution or linear combinations.

I can write two equations:

1 equation about the total number of people.
You have 7 per van + 25 per bus = 231 (Total)
The equation: 7x + 25y = 231

2nd equation is about the number of vehicles.
number of vans + number of buses = 15 (Total)
Equation: x + y = 15

So our system of equations is:
7x + 25y = 231
x + y = 15

I am going to solve by using substitution.

First I will solve x + y = 15 for y:

x -x + y = 15 - x Subtract x from both sides.
y = 15 - x

Now substitute y = 15 -x into the first equation:

7x + 25y = 231
7x + 25(15 -x) = 231 Substitute
7x + 375 - 25x = 231 Distribute
7x - 25x + 375 = 231 Write like terms together
-18x + 375 = 231 Combine like terms
-18x + 375 - 375 = 231 - 375 Subtract 375 from both sides

-18x = -144 Simplify
-18x/-18 = -144/-18 Divide -18 from both sides
x = 8

The number of vans equals 8.

Use x + y = 15 to find the number of buses.
8 + y = 15
8 + 7 = 15
Therefore, there were 7 buses.

Check:
7(8) +25(7) = 231
231 = 231

So, there were 8 vans and 7 buses.

Hope this helps,
Karin

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