Set up and solve the system of linear equations using matrices.

Question

On July 7, 2004, Shania Twain was scheduled to perform at the TD Waterhouse Centre in Orlando, Florida. The center offered 18,039 seats for the concert in three seating classifications: floor, lower, and upper. Based upon their location, tickets were offered at three different prices: $80, $65, and $45.† Suppose the average price of a floor ticket was $80, the average price of a lower ticket was $65, and the average price of an upper ticket was $45. If concert planners expected a total of 13,000 tickets would be sold, including 5 times as many tickets on the lower level as on the floor, how many of each type of ticket would have to be sold to reach $787,500 in ticket revenue?

 

floor tickets= 

lower tickets=

upper tickets=

Answer 1

Lets dive right in, let F be number of floor tickets, L be number of lower tickets, and U be number of upper tickets sold.

 

The concert planners expected 13,000 tickets to be sold. From this sentence we can infer that F + L + U = 13,000, since F+L+U is total number of tickets.

 

The planners expect to earn $787,500. In order to make this into an equation we also need the ticket prices, since the price times the number of tickets sold will give us how much money was made on each ticket, we then add it up to get expected earnings. All of this together gives us the equation:

80F + 65L + 45U = 787,500.

 

Now the last part is a little tricky. The planners expect 5 times as many lower tickets as floor tickets, which gives us the equation 5F = L. But this is not in the form we want, so bring the L over and add by 0*U. We get:

5F - L + 0*U = 0

 

All together we get:

F + L + U = 13,000

80F + 65L + 45U = 787,500

5F - L + 0*U = 0

 

Now we can represent this as a matrix vector product:

1    1   1           F              13,000

80  65 45    *    L       =    787,500

5  -1   0            U             0

 

We can now solve this by inverting the matrix.

 

I plugged this into Matlab and got:

F=1500

L=7500

U= 4000

 

Hope this helps, if you want me to go over how to invert a matrix let me know and I'll answer again.