Having a hard time figuring out the formula

Question

A class used cars and vans to go on a field trip because all the buses were already in use. They used eight vehicles to go on the trip. Each car holds three students in each van holds 12 students. If 51 students went on the trip, then how many of each type of vehicle did the class use?

Answer 1

c + v = 8 —— Equation 1

3c + 12v = 51 ——Equation 2

Lets multiply Equation 1 by the number 3 so we can have the same number in front of c for both equations. We will get

3c + 3v = 24 —— New Equation

Now let’s subtract the new equation from equation 2

3c + 12v = 51

—

3c + 3v = 24

————————

0c + 9v = 27

We now have 9v = 27

We divide both sides by 9

v= 3

now we substitute v = 3 into equation 1

c + 3 = 8

Then subtract 3 from both sides of the equation

c + 3 - 3 = 8 - 3

c = 5

So the class used 5 cars and 3 vans

Answer 2

c + v = 8

3c + 12v = 51

First, solve the first equation for c or v. I'll do c.

c = 8 - v

Now, substitute 8 - v for c in the 2nd equation, since they are equal.

3(8 - v) + 12v = 51

24 - 3v + 12v = 51

9v = 27

v = 3

c = 8 - 3

c = 5