First, your goal is to find the MINIMUM amount to give Keith. So any max becomes irrelevant because it would go over the minimum amount. In order to guarantee Keith a portion of the revenue, you have to find the total amount of each tickets sold. Let us represent the amount of $15 ticket as x and the amount of $25 ticket as y. Of the $15 tickets you give $5 and of the $25 tickets you give $7.5.
So you make an equation 5x + 7.5y =?, you cant make a graph off that, so you have to find that final value.
Your first step becomes to find the total minimum of people attending and the amount of people buying each ticket.
You have a total of 12,500 attending, so to tell the guaranteed amount of attendees,
you have x + y = 12,500.
You also have $225,000 from those tickets. So to get the total amount in dollars I multiply the amount of tickets by the cost of the tickets. So, 15x +25y = 225,000.
As you realize, you have similar variables. So you start to compare them.
x + y = 12,500.
15x +25y = 225,000.
When you graph this, the solution of the two lines will be wherever they intersect. However, since the large numbers make it difficult to see the exact answer, I am going to do some algebra. I am going to use elimination in this case to isolate a single variable. So, by choice, I want to get rid of x. To do it, I have to multiply by a factor of –15
-15x + -15y = -187,500.
15x +25y = 225,000.
0+10y = 37500
*party sounds*, we found the first variable. However, we still need to find x. So plug it in any of the equations since both equations are true.
I find x + y = 12,500 simple, so I do x+3750 = 12500.
note: you can also get the same answer in the graph from the intersect between the two lines.
Remember that Keith guy? You still have to tell him how much money he is guaranteed. So bringing down the first equation you made.
5x + 7.5y =?
Substitute your new x and y variables, and solve
5*8750 + 7.5*3750 = 71875
You can guarantee Keith at least $71,875 for doing the concert.