SOLUTION: Adult tickets for a play cost $8 and child tickets cost $6. If there were 25 people at a performance and the theater collected $188 from ticket sales, how many adults and how many

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Question 112100: Adult tickets for a play cost $8 and child tickets cost $6. If there were 25 people at a performance and the theater collected $188 from ticket sales, how many adults and 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 adults, y=# of children

Translate the following word problem:
x%2By=25 "If there were 25 people at a performance"

8x%2B6y=188 "Adult tickets for a play cost $8 and child tickets cost $6...the theater collected $188 from ticket sales"


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=25
8%2Ax%2B6%2Ay=188

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=25-1%2AxSubtract 1%2Ax from both sides

y=%2825-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=25-1%2Ax Now we've fully isolated y

Since y equals 25-1%2Ax we can substitute the expression 25-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


8%2Ax%2B6%2Ahighlight%28%2825-1%2Ax%29%29=188 Replace y with 25-1%2Ax. Since this eliminates y, we can now solve for x.

8%2Ax%2B6%2A%2825%29%2B6%28-1%29x=188 Distribute 6 to 25-1%2Ax

8%2Ax%2B150-6%2Ax=188 Multiply



8%2Ax%2B150-6%2Ax=188 Reduce any fractions

8%2Ax-6%2Ax=188-150 Subtract 150 from both sides


8%2Ax-6%2Ax=38 Combine the terms on the right side



2%2Ax=38 Now combine the terms on the left side.


cross%28%281%2F2%29%282%2F1%29%29x=%2838%2F1%29%281%2F2%29 Multiply both sides by 1%2F2. This will cancel out 2%2F1 and isolate x

So when we multiply 38%2F1 and 1%2F2 (and simplify) we get



x=19 <---------------------------------One answer

Now that we know that x=19, lets substitute that in for x to solve for y

8%2819%29%2B6%2Ay=188 Plug in x=19 into the 2nd equation

152%2B6%2Ay=188 Multiply

6%2Ay=188-152Subtract 152 from both sides

6%2Ay=36 Combine the terms on the right side

cross%28%281%2F6%29%286%29%29%2Ay=%2836%2F1%29%281%2F6%29 Multiply both sides by 1%2F6. This will cancel out 6 on the left side.

y=36%2F6 Multiply the terms on the right side


y=6 Reduce


So this is the other answer


y=6<---------------------------------Other answer


So our solution is

x=19 and y=6

which can also look like

(19,6)

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

1%2Ax%2B1%2Ay=25
8%2Ax%2B6%2Ay=188

we get


graph of 1%2Ax%2B1%2Ay=25 (red) and 8%2Ax%2B6%2Ay=188 (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 (19,6). This verifies our answer.


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Check:

Plug in (19,6) into the system of equations


Let x=19 and y=6. Now plug those values into the equation 1%2Ax%2B1%2Ay=25

1%2A%2819%29%2B1%2A%286%29=25 Plug in x=19 and y=6


19%2B6=25 Multiply


25=25 Add


25=25 Reduce. Since this equation is true the solution works.


So the solution (19,6) satisfies 1%2Ax%2B1%2Ay=25



Let x=19 and y=6. Now plug those values into the equation 8%2Ax%2B6%2Ay=188

8%2A%2819%29%2B6%2A%286%29=188 Plug in x=19 and y=6


152%2B36=188 Multiply


188=188 Add


188=188 Reduce. Since this equation is true the solution works.


So the solution (19,6) satisfies 8%2Ax%2B6%2Ay=188


Since the solution (19,6) satisfies the system of equations


1%2Ax%2B1%2Ay=25
8%2Ax%2B6%2Ay=188


this verifies our answer.





So there were 19 adults and 6 children