SOLUTION: Adult tickets for a play cost $20 and child tickets cost $16. If there were 27 people at a performance and the theater collected $464 from ticket sales, how many adults and how ma

Algebra ->  Customizable Word Problem Solvers  -> Misc -> SOLUTION: Adult tickets for a play cost $20 and child tickets cost $16. If there were 27 people at a performance and the theater collected $464 from ticket sales, how many adults and how ma      Log On

Ad: Over 600 Algebra Word Problems at edhelper.com


    

Question 92529: Adult tickets for a play cost $20 and child tickets cost $16. If there were 27 people at a performance and the theater collected $464 from ticket sales, how many adults and how many children attended the play?
A) 10 adults and 21 children
B) 9 adults and 20 children
C) 8 adults and 19 children
D) 7 adults and 18 children
Thank you for any help...
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 word problem into a system of equations:
20x%2B16y=464 the cost of the adults' tickets plus the cost of the children's tickets is equal to $464
x%2By=27 the sum of all who attended is 27

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


Lets start with the given system of linear equations

20%2Ax%2B16%2Ay=464
1%2Ax%2B1%2Ay=27

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

16%2Ay=464-20%2AxSubtract 20%2Ax from both sides

y=%28464-20%2Ax%29%2F16 Divide both sides by 16.


Which breaks down and reduces to



y=29-%285%2F4%29%2Ax Now we've fully isolated y

Since y equals 29-%285%2F4%29%2Ax we can substitute the expression 29-%285%2F4%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%2829-%285%2F4%29%2Ax%29%29=27 Replace y with 29-%285%2F4%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%2829%29%2B1%28-5%2F4%29x=27 Distribute 1 to 29-%285%2F4%29%2Ax

1%2Ax%2B29-%285%2F4%29%2Ax=27 Multiply



1%2Ax%2B29-%285%2F4%29%2Ax=27 Reduce any fractions

1%2Ax-%285%2F4%29%2Ax=27-29 Subtract 29 from both sides


1%2Ax-%285%2F4%29%2Ax=-2 Combine the terms on the right side



%284%2F4%29%2Ax-%285%2F4%29x=-2 Make 1 into a fraction with a denominator of 4

%28-1%2F4%29%2Ax=-2 Now combine the terms on the left side.


cross%28%284%2F-1%29%28-1%2F4%29%29x=%28-2%2F1%29%284%2F-1%29 Multiply both sides by 4%2F-1. This will cancel out -1%2F4 and isolate x

So when we multiply -2%2F1 and 4%2F-1 (and simplify) we get



x=8 <---------------------------------One answer

Now that we know that x=8, lets substitute that in for x to solve for y

1%288%29%2B1%2Ay=27 Plug in x=8 into the 2nd equation

8%2B1%2Ay=27 Multiply

1%2Ay=27-8Subtract 8 from both sides

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

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

y=19%2F1 Multiply the terms on the right side


y=19 Reduce


So this is the other answer


y=19<---------------------------------Other answer


So our solution is

x=8 and y=19

which can also look like

(8,19)

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

20%2Ax%2B16%2Ay=464
1%2Ax%2B1%2Ay=27

we get


graph of 20%2Ax%2B16%2Ay=464 (red) and 1%2Ax%2B1%2Ay=27 (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 (8,19). This verifies our answer.


-----------------------------------------------------------------------------------------------
Check:

Plug in (8,19) into the system of equations


Let x=8 and y=19. Now plug those values into the equation 20%2Ax%2B16%2Ay=464

20%2A%288%29%2B16%2A%2819%29=464 Plug in x=8 and y=19


160%2B304=464 Multiply


464=464 Add


464=464 Reduce. Since this equation is true the solution works.


So the solution (8,19) satisfies 20%2Ax%2B16%2Ay=464



Let x=8 and y=19. Now plug those values into the equation 1%2Ax%2B1%2Ay=27

1%2A%288%29%2B1%2A%2819%29=27 Plug in x=8 and y=19


8%2B19=27 Multiply


27=27 Add


27=27 Reduce. Since this equation is true the solution works.


So the solution (8,19) satisfies 1%2Ax%2B1%2Ay=27


Since the solution (8,19) satisfies the system of equations


20%2Ax%2B16%2Ay=464
1%2Ax%2B1%2Ay=27


this verifies our answer.





So 8 adults and 19 children attended. This means the answer is C)