SOLUTION: The sophomore class at south high school raised $800 from sales tickets to a dance. tickets sold for $1.50 in advance and $ 2.00 at the door. if a total of 475 tickets were sold, w

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Question 114793: The sophomore class at south high school raised $800 from sales tickets to a dance. tickets sold for $1.50 in advance and $ 2.00 at the door. if a total of 475 tickets were sold, what were the number of tickets sold at the door
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let x=# of advance tickets, y=# of door tickets
Since there were a total of 475 tickets sold, the first equation is

x%2By=475


And since those tickets brought in $800, the next equation is


1.5x%2B2y=800

Now multiply both sides by 10 to get

15x%2B20y=8000



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=475
15%2Ax%2B20%2Ay=8000

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=475-1%2AxSubtract 1%2Ax from both sides

y=%28475-1%2Ax%29 Divide both sides by 1.


Which breaks down and reduces to



y=475-1%2Ax Now we've fully isolated y

Since y equals 475-1%2Ax we can substitute the expression 475-1%2Ax into y of the 2nd equation. This will eliminate y so we can solve for x.


15%2Ax%2B20%2Ahighlight%28%28475-1%2Ax%29%29=8000 Replace y with 475-1%2Ax. Since this eliminates y, we can now solve for x.

15%2Ax%2B20%2A%28475%29%2B20%28-1%29x=8000 Distribute 20 to 475-1%2Ax

15%2Ax%2B9500-20%2Ax=8000 Multiply



15%2Ax%2B9500-20%2Ax=8000 Reduce any fractions

15%2Ax-20%2Ax=8000-9500 Subtract 9500 from both sides


15%2Ax-20%2Ax=-1500 Combine the terms on the right side



-5%2Ax=-1500 Now combine the terms on the left side.


cross%28%281%2F-5%29%28-5%2F1%29%29x=%28-1500%2F1%29%281%2F-5%29 Multiply both sides by 1%2F-5. This will cancel out -5%2F1 and isolate x

So when we multiply -1500%2F1 and 1%2F-5 (and simplify) we get



x=300 <---------------------------------One answer

Now that we know that x=300, lets substitute that in for x to solve for y

15%28300%29%2B20%2Ay=8000 Plug in x=300 into the 2nd equation

4500%2B20%2Ay=8000 Multiply

20%2Ay=8000-4500Subtract 4500 from both sides

20%2Ay=3500 Combine the terms on the right side

cross%28%281%2F20%29%2820%29%29%2Ay=%283500%2F1%29%281%2F20%29 Multiply both sides by 1%2F20. This will cancel out 20 on the left side.

y=3500%2F20 Multiply the terms on the right side


y=175 Reduce


So this is the other answer


y=175<---------------------------------Other answer


So our solution is

x=300 and y=175

which can also look like

(300,175)

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

1%2Ax%2B1%2Ay=475
15%2Ax%2B20%2Ay=8000

we get


graph of 1%2Ax%2B1%2Ay=475 (red) and 15%2Ax%2B20%2Ay=8000 (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 (300,175). This verifies our answer.


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Check:

Plug in (300,175) into the system of equations


Let x=300 and y=175. Now plug those values into the equation 1%2Ax%2B1%2Ay=475

1%2A%28300%29%2B1%2A%28175%29=475 Plug in x=300 and y=175


300%2B175=475 Multiply


475=475 Add


475=475 Reduce. Since this equation is true the solution works.


So the solution (300,175) satisfies 1%2Ax%2B1%2Ay=475



Let x=300 and y=175. Now plug those values into the equation 15%2Ax%2B20%2Ay=8000

15%2A%28300%29%2B20%2A%28175%29=8000 Plug in x=300 and y=175


4500%2B3500=8000 Multiply


8000=8000 Add


8000=8000 Reduce. Since this equation is true the solution works.


So the solution (300,175) satisfies 15%2Ax%2B20%2Ay=8000


Since the solution (300,175) satisfies the system of equations


1%2Ax%2B1%2Ay=475
15%2Ax%2B20%2Ay=8000


this verifies our answer.






So they sold 300 tickets in advance and 175 tickets at the door.