Question 145298
Let x=# of orchestra seats and y=# of balcony seats


Since the "students sold 650 tickets", this means that {{{x+y=650}}}. This is our first equation.



Also, since the total cost of x orchestra seats (at $7.50) and y balcony seats (at $3.50) is $4,375.00, this means that {{{7.5x+3.5y=4375}}}. This is our second equation.



{{{10(7.5x+3.5y)=10(4375)}}} Multiply both sides of the second equation by 10. This will make every number a whole number.



{{{75x+35y=43750}}} Distribute and multiply.







So we now have this system of equations:


{{{system(x+y=650,75x+35y=43750)}}}



Let's use substitution to solve this system.


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





So let's isolate y in the first equation


{{{x+y=650}}} Start with the first equation



{{{y=650-x}}}  Subtract {{{x}}} from both sides



{{{y=-x+650}}} Rearrange the equation



---------------------


Since {{{y=-x+650}}}, we can now replace each {{{y}}} in the second equation with {{{-x+650}}} to solve for {{{x}}}




{{{75x+35highlight((-x+650))=43750}}} Plug in {{{y=-x+650}}} into the first equation. In other words, replace each {{{y}}} with {{{-x+650}}}. Notice we've eliminated the {{{y}}} variables. So we now have a simple equation with one unknown.




{{{75x+(35)(-1)x+(35)(650)=43750}}} Distribute {{{35}}} to {{{-x+650}}}



{{{75x-35x+22750=43750}}} Multiply



{{{40x+22750=43750}}} Combine like terms on the left side



{{{40x=43750-22750}}}Subtract 22750 from both sides



{{{40x=21000}}} Combine like terms on the right side



{{{x=(21000)/(40)}}} Divide both sides by 40 to isolate x




{{{x=525}}} Divide






-----------------First Answer------------------------------



So the first part of our answer is: {{{x=525}}}










Since we know that {{{x=525}}} we can plug it into the equation {{{y=-x+650}}} (remember we previously solved for {{{y}}} in the first equation).




{{{y=-x+650}}} Start with the equation where {{{y}}} was previously isolated.



{{{y=-(525)+650}}} Plug in {{{x=525}}}



{{{y=-525+650}}} Multiply



{{{y=125}}} Combine like terms 




-----------------Second Answer------------------------------



So the second part of our answer is: {{{y=125}}}










-----------------Summary------------------------------


So our answers are:


{{{x=525}}} and {{{y=125}}}




This means that the students sold 525 orchestra seats and 125 balcony seats