SOLUTION: Adult tickets for a play cost $9 and child tickets cost $8. If there were 23 people at a performance and the theater collected $193 from ticket sales, how many children attended t

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Question 97035: Adult tickets for a play cost $9 and child tickets cost $8. If there were 23 people at a performance and the theater collected $193 from ticket sales, how many children attended the play?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=# of children, y=# of adults


Set up the following system
x%2By=23 "If there were 23 people at a performance"
8x%2B9y=193 "Adult tickets for a play cost $9 and child tickets cost $8"


Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

1%2Ax%2B1%2Ay=23
8%2Ax%2B9%2Ay=193

Now in order to solve this system by using substitution, we need to solve (or isolate) one variable. I'm going to choose y.

Solve for y for the first equation

1%2Ay=23-1%2AxSubtract 1%2Ax from both sides

y=%2823-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=23-1%2Ax Now we've fully isolated y

Since y equals 23-1%2Ax we can substitute the expression 23-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


8%2Ax%2B9%2Ahighlight%28%2823-1%2Ax%29%29=193 Replace y with 23-1%2Ax. Since this eliminates y, we can now solve for x.

8%2Ax%2B9%2A%2823%29%2B9%28-1%29x=193 Distribute 9 to 23-1%2Ax

8%2Ax%2B207-9%2Ax=193 Multiply



8%2Ax%2B207-9%2Ax=193 Reduce any fractions

8%2Ax-9%2Ax=193-207 Subtract 207 from both sides


8%2Ax-9%2Ax=-14 Combine the terms on the right side



-1%2Ax=-14 Now combine the terms on the left side.


cross%28%281%2F-1%29%28-1%2F1%29%29x=%28-14%2F1%29%281%2F-1%29 Multiply both sides by 1%2F-1. This will cancel out -1%2F1 and isolate x

So when we multiply -14%2F1 and 1%2F-1 (and simplify) we get



x=14 <---------------------------------One answer

Now that we know that x=14, lets substitute that in for x to solve for y

8%2814%29%2B9%2Ay=193 Plug in x=14 into the 2nd equation

112%2B9%2Ay=193 Multiply

9%2Ay=193-112Subtract 112 from both sides

9%2Ay=81 Combine the terms on the right side

cross%28%281%2F9%29%289%29%29%2Ay=%2881%2F1%29%281%2F9%29 Multiply both sides by 1%2F9. This will cancel out 9 on the left side.

y=81%2F9 Multiply the terms on the right side


y=9 Reduce


So this is the other answer


y=9<---------------------------------Other answer


So our solution is

x=14 and y=9

which can also look like

(14,9)

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

1%2Ax%2B1%2Ay=23
8%2Ax%2B9%2Ay=193

we get


graph of 1%2Ax%2B1%2Ay=23 (red) and 8%2Ax%2B9%2Ay=193 (green) (hint: you may have to solve for y to graph these) intersecting at the blue circle.


and we can see that the two equations intersect at (14,9). This verifies our answer.


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Check:

Plug in (14,9) into the system of equations


Let x=14 and y=9. Now plug those values into the equation 1%2Ax%2B1%2Ay=23

1%2A%2814%29%2B1%2A%289%29=23 Plug in x=14 and y=9


14%2B9=23 Multiply


23=23 Add


23=23 Reduce. Since this equation is true the solution works.


So the solution (14,9) satisfies 1%2Ax%2B1%2Ay=23



Let x=14 and y=9. Now plug those values into the equation 8%2Ax%2B9%2Ay=193

8%2A%2814%29%2B9%2A%289%29=193 Plug in x=14 and y=9


112%2B81=193 Multiply


193=193 Add


193=193 Reduce. Since this equation is true the solution works.


So the solution (14,9) satisfies 8%2Ax%2B9%2Ay=193


Since the solution (14,9) satisfies the system of equations


1%2Ax%2B1%2Ay=23
8%2Ax%2B9%2Ay=193


this verifies our answer.






So that means 14 children and 9 adults attended the play