matrix equations

Question

An outdoor amphitheater has 25,000 seats. Ticket prices are $15, $17, and $25 and the number of tickets priced at $17 must equal two times the number priced at $25 How many tickets of each type should be sold (assuming all seats can be sold) to bring in $434,500? Set up a system of equations and solve using inverse matrix method.

a) Let x = _____________________________________

y = _____________________________________

z = _____________________________________

b) Write the equations associated with the above problem:

x+y+z=25000
15x+17y+25z=434500
y=2z




c) Write the matrix equation associated with the above problem:






d) The solution is (use matrix algebra):

Number of $15 tickets: 11,179

Number of $17 tickets: 9,214

Number of $25 tickets: 4,607



e) If you change the ticket prices to $8, $12, and $20 and now must bring in $256,000, the new solution is:

Number of $8 tickets: ____________________

Number of $12 tickets: _________________

Number of $20 tickets: ________________

Answer 1

Let your matrtix A be

 

1   1   1

 

15 17 25

 

0  1   -2

 

Let w be the column vector (x,y,z) and let b be the column vector (25000, 434500, 0). Then the matrix equation is Aw=b. To solve for w multiply both sides by A^-1 on the left to get w=A^-1 b.

 

I hope it helps.