Billy went to the show and spent $78 on a combined total of 9 items including rides and showbags. if each showbag cost $12 and each ride cost $7, how many of each did billy buy?

First, set x for showbags and y for rides. 
So now we need to make a system equations. 
Since he bought 9 items, x+y=9. 
And because he spent $78 and we know the price of each item, the second equation is 12x+7y=78.

Now solve for the system of equations by elimination. 
12x+7y=78
-7(x+y=9) 

Now the new equations are:
12x+7y=78
-7x-7y=-63

Now just subtract. 
5x=15

x=3. 

And since originally we made x represent showbags, he bought 3 showbags. Which means he bought 6 rides because 9-3=6. 

If you wanna check your answer, just plug in. 

x+y=9
3+6=9

12(3)+6(7)=78
36+42=78

Billy bought 3 showbags and 6 rides.