SOLUTION: Adult tickets for a play cost $11 and the child tickets cost $10. If there were 23 people at a performance and th theater collected $246 from tickets sales, how many children atten

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: Adult tickets for a play cost $11 and the child tickets cost $10. If there were 23 people at a performance and th theater collected $246 from tickets sales, how many children atten      Log On


   

Question 88484: Adult tickets for a play cost $11 and the child tickets cost $10. If there were 23 people at a performance and th theater collected $246 from tickets 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 adults and y=# of children
Solved by pluggable solver: Solving a linear system of equations by subsitution


Lets start with the given system of linear equations

11%2Ax%2B10%2Ay=246
1%2Ax%2B1%2Ay=23

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

10%2Ay=246-11%2AxSubtract 11%2Ax from both sides

y=%28246-11%2Ax%29%2F10 Divide both sides by 10.


Which breaks down and reduces to



y=123%2F5-%2811%2F10%29%2Ax Now we've fully isolated y

Since y equals 123%2F5-%2811%2F10%29%2Ax we can substitute the expression 123%2F5-%2811%2F10%29%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


1%2Ax%2B1%2Ahighlight%28%28123%2F5-%2811%2F10%29%2Ax%29%29=23 Replace y with 123%2F5-%2811%2F10%29%2Ax. Since this eliminates y, we can now solve for x.

1%2Ax%2B1%2A%28123%2F5%29%2B1%28-11%2F10%29x=23 Distribute 1 to 123%2F5-%2811%2F10%29%2Ax

1%2Ax%2B123%2F5-%2811%2F10%29%2Ax=23 Multiply



1%2Ax%2B123%2F5-%2811%2F10%29%2Ax=23 Reduce any fractions

1%2Ax-%2811%2F10%29%2Ax=23-123%2F5 Subtract 123%2F5 from both sides


1%2Ax-%2811%2F10%29%2Ax=115%2F5-123%2F5 Make 23 into a fraction with a denominator of 5


1%2Ax-%2811%2F10%29%2Ax=-8%2F5 Combine the terms on the right side



%2810%2F10%29%2Ax-%2811%2F10%29x=-8%2F5 Make 1 into a fraction with a denominator of 10

%28-1%2F10%29%2Ax=-8%2F5 Now combine the terms on the left side.


cross%28%2810%2F-1%29%28-1%2F10%29%29x=%28-8%2F5%29%2810%2F-1%29 Multiply both sides by 10%2F-1. This will cancel out -1%2F10 and isolate x

So when we multiply -8%2F5 and 10%2F-1 (and simplify) we get



x=16 <---------------------------------One answer

Now that we know that x=16, lets substitute that in for x to solve for y

1%2816%29%2B1%2Ay=23 Plug in x=16 into the 2nd equation

16%2B1%2Ay=23 Multiply

1%2Ay=23-16Subtract 16 from both sides

1%2Ay=7 Combine the terms on the right side

cross%28%281%2F1%29%281%29%29%2Ay=%287%2F1%29%281%2F1%29 Multiply both sides by 1%2F1. This will cancel out 1 on the left side.

y=7%2F1 Multiply the terms on the right side


y=7 Reduce


So this is the other answer


y=7<---------------------------------Other answer


So our solution is

x=16 and y=7

which can also look like

(16,7)

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

11%2Ax%2B10%2Ay=246
1%2Ax%2B1%2Ay=23

we get


graph of 11%2Ax%2B10%2Ay=246 (red) and 1%2Ax%2B1%2Ay=23 (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 (16,7). This verifies our answer.


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Check:

Plug in (16,7) into the system of equations


Let x=16 and y=7. Now plug those values into the equation 11%2Ax%2B10%2Ay=246

11%2A%2816%29%2B10%2A%287%29=246 Plug in x=16 and y=7


176%2B70=246 Multiply


246=246 Add


246=246 Reduce. Since this equation is true the solution works.


So the solution (16,7) satisfies 11%2Ax%2B10%2Ay=246



Let x=16 and y=7. Now plug those values into the equation 1%2Ax%2B1%2Ay=23

1%2A%2816%29%2B1%2A%287%29=23 Plug in x=16 and y=7


16%2B7=23 Multiply


23=23 Add


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


So the solution (16,7) satisfies 1%2Ax%2B1%2Ay=23


Since the solution (16,7) satisfies the system of equations


11%2Ax%2B10%2Ay=246
1%2Ax%2B1%2Ay=23


this verifies our answer.





So the number of adults that attended is 16 and the number of children is 7