SOLUTION: I need help on this word problem. Thank You in advance for your help. Adult tickets for a play cost $8 and child tickets cost $1. If there were 22 people at a performance and th

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Question 112249: I need help on this word problem. Thank You in advance for your help.
Adult tickets for a play cost $8 and child tickets cost $1. If there were 22 people at a performance and the theater collected $50, 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 adult tickets, y=# of child tickets

Translate the following word problem:

"If there were 22 people at a performance"----> x%2By=22

"dult tickets for a play cost $8 and child tickets cost $1...the theater collected $50"---->8x%2By=50

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=22
8%2Ax%2B1%2Ay=50

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=22-1%2AxSubtract 1%2Ax from both sides

y=%2822-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=22-1%2Ax Now we've fully isolated y

Since y equals 22-1%2Ax we can substitute the expression 22-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


8%2Ax%2B1%2Ahighlight%28%2822-1%2Ax%29%29=50 Replace y with 22-1%2Ax. Since this eliminates y, we can now solve for x.

8%2Ax%2B1%2A%2822%29%2B1%28-1%29x=50 Distribute 1 to 22-1%2Ax

8%2Ax%2B22-1%2Ax=50 Multiply



8%2Ax%2B22-1%2Ax=50 Reduce any fractions

8%2Ax-1%2Ax=50-22 Subtract 22 from both sides


8%2Ax-1%2Ax=28 Combine the terms on the right side



7%2Ax=28 Now combine the terms on the left side.


cross%28%281%2F7%29%287%2F1%29%29x=%2828%2F1%29%281%2F7%29 Multiply both sides by 1%2F7. This will cancel out 7%2F1 and isolate x

So when we multiply 28%2F1 and 1%2F7 (and simplify) we get



x=4 <---------------------------------One answer

Now that we know that x=4, lets substitute that in for x to solve for y

8%284%29%2B1%2Ay=50 Plug in x=4 into the 2nd equation

32%2B1%2Ay=50 Multiply

1%2Ay=50-32Subtract 32 from both sides

1%2Ay=18 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%2818%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=18%2F1 Multiply the terms on the right side


y=18 Reduce


So this is the other answer


y=18<---------------------------------Other answer


So our solution is

x=4 and y=18

which can also look like

(4,18)

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

1%2Ax%2B1%2Ay=22
8%2Ax%2B1%2Ay=50

we get


graph of 1%2Ax%2B1%2Ay=22 (red) and 8%2Ax%2B1%2Ay=50 (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 (4,18). This verifies our answer.


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Check:

Plug in (4,18) into the system of equations


Let x=4 and y=18. Now plug those values into the equation 1%2Ax%2B1%2Ay=22

1%2A%284%29%2B1%2A%2818%29=22 Plug in x=4 and y=18


4%2B18=22 Multiply


22=22 Add


22=22 Reduce. Since this equation is true the solution works.


So the solution (4,18) satisfies 1%2Ax%2B1%2Ay=22



Let x=4 and y=18. Now plug those values into the equation 8%2Ax%2B1%2Ay=50

8%2A%284%29%2B1%2A%2818%29=50 Plug in x=4 and y=18


32%2B18=50 Multiply


50=50 Add


50=50 Reduce. Since this equation is true the solution works.


So the solution (4,18) satisfies 8%2Ax%2B1%2Ay=50


Since the solution (4,18) satisfies the system of equations


1%2Ax%2B1%2Ay=22
8%2Ax%2B1%2Ay=50


this verifies our answer.





So there were 4 adults and 18 children