SOLUTION: I am working on this problem. I am trying to set it up correctly. Adult tickets to a play cost $13 and child tickets cost $10. If there were 22 people at a performance and the th

Algebra ->  Customizable Word Problem Solvers  -> Numbers -> SOLUTION: I am working on this problem. I am trying to set it up correctly. Adult tickets to a play cost $13 and child tickets cost $10. If there were 22 people at a performance and the th      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 111859: I am working on this problem. I am trying to set it up correctly.
Adult tickets to a play cost $13 and child tickets cost $10. If there were 22 people at a performance and the theater collected $277 from ticket sales, how many adults and how many children attended the play.
First,
x= Adult
y = child
x+y=22
13x + 10y=22
I am not certain if I should multiply the second equation by -10 so that I can use the addtion method.
What is the best way to solve this problem?
Thank you,
Tracy


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
There's one little problem: the second equation 13x+%2B+10y=22 represents the total sales (the left side is the sum of the amount of money made and the right side is the total money). So the second equation should be
13x+%2B+10y=277


Solved by pluggable solver: Solving a System of Linear Equations by Elimination/Addition


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=22
13%2Ax%2B10%2Ay=277

In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).

So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.

So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 13 to some equal number, we could try to get them to the LCM.

Since the LCM of 1 and 13 is 13, we need to multiply both sides of the top equation by 13 and multiply both sides of the bottom equation by -1 like this:

13%2A%281%2Ax%2B1%2Ay%29=%2822%29%2A13 Multiply the top equation (both sides) by 13
-1%2A%2813%2Ax%2B10%2Ay%29=%28277%29%2A-1 Multiply the bottom equation (both sides) by -1


So after multiplying we get this:
13%2Ax%2B13%2Ay=286
-13%2Ax-10%2Ay=-277

Notice how 13 and -13 add to zero (ie 13%2B-13=0)


Now add the equations together. In order to add 2 equations, group like terms and combine them
%2813%2Ax-13%2Ax%29%2B%2813%2Ay-10%2Ay%29=286-277

%2813-13%29%2Ax%2B%2813-10%29y=286-277

cross%2813%2B-13%29%2Ax%2B%2813-10%29%2Ay=286-277 Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.



So after adding and canceling out the x terms we're left with:

3%2Ay=9

y=9%2F3 Divide both sides by 3 to solve for y



y=3 Reduce


Now plug this answer into the top equation 1%2Ax%2B1%2Ay=22 to solve for x

1%2Ax%2B1%283%29=22 Plug in y=3


1%2Ax%2B3=22 Multiply



1%2Ax=22-3 Subtract 3 from both sides

1%2Ax=19 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ax=%2819%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.


x=19 Multiply the terms on the right side


So our answer is

x=19, y=3

which also looks like

(19, 3)

Notice if we graph the equations (if you need help with graphing, check out this solver)

1%2Ax%2B1%2Ay=22
13%2Ax%2B10%2Ay=277

we get



graph of 1%2Ax%2B1%2Ay=22 (red) 13%2Ax%2B10%2Ay=277 (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).


and we can see that the two equations intersect at (19,3). This verifies our answer.


So this means there were 19 adults and 3 children at the play