How many students can a van carry? How many students can a bus carry?

Question

Yellowstone National Park 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 12 buses with 493 students. High School B rented and filled 8 vans and 2 buses with 204 students. Every van had the same number of students in it as did the buses. How many students can a van carry? How many students can a bus carry?

We need to find out what x and y represents and the system of equations.

Answer 1

Let "v" be the number of students in a van.

Let "b" be the number of students in a bus.

For High School A

5v + 12b = 493

For High School B

8v + 2b = 204

I prefer to use Elimination. To do so, multiply the second equation be -6 to get:

-48v - 12b = -1224

Add it to the first equation:

5v + 12b = 493

-48v - 12b = -1224

--------------------------

-43v = -731

v = 17

Using the first equation and plugging in v = 17 we get:

5v + 12b = 493

5(17) + 12b = 493

85 + 12b = 493

12b = 408

b = 34

So, 12 students in each van and 34 students in each bus.

Answer 2

System of equations: Where x = vans, y = buses

5x + 12y = 493

8x + 2y = 204

Lets solve the first equation for y.

y = (493 - 5x) / 12

We can then substitute this y for the y in the second equation.

8x + 2( ( 493 - 5x) / 12) = 204

Lets make this more readable;

8x + 2(493/12) - 2(5x/12) = 204

Lets start isolating x;

8x - 2(5x/12) = 204 - 2(493/12)

8x - 10x/12 = 204 - 986/12

Lets get rid of those pesky 12s by multiplying each side by 12 to get:

96x - 10x = 2448 - 986

86x = 1462

Divide both sides by 86;

x = 17

Now we know that the vans carry 17 students.

Lets plug that 17 into x for the first equation to solve for y (buses).

5(17) + 12y = 493

85 + 12y = 493

12y = 408

y = 34

We now know that the buses can carry 34 students.

x = 17, y = 34

You can then check the second equation with this values to make sure that you get 204.

Another method you can use to solve this problem is to isolate for y in both equations then plot it on a graph. Once both equations are plotted you can determine were they intersect (17, 34) or (x, y).