Question

The Pack Theatre has 690 seats. For the upcoming production of Rock of Ages the auditorium has sectioned off the seats into three areas. Seats sell for $20 in section A, $15 in section B, and $10 in section C. The number of seats in section A equals 328 seats less than the number of seats in sections B and C combined. Suppose the stadium takes in $10,030 from each sold-out event.

Driving Question: How can I write and solve a system of equations to find the number of seats in each section?

Write and solve a system of equations to find the number of seats in each section.
Clearly define your variables
Write the equations
State your strategy for solving and show your work
Write your answer in a complete sentence

Answer 1

(181,264,245) or x=181 y=264 z=245

Explanation:

a) Clearly define your variables.

x=section a

y=section b

z=section c

b) Write a system of equations.

x+y+z=690

20x+15y+10z=10,030

x+328=y+z ---> x-y-z=-328

c) We can solve using a matrix.

The numbers in parenthesis are what go in the matrix.

x(1)+y(1)+z(1)=690(690)

20x(20)+15x(15)+10x(10)=10,030(10,030)

x(1)-y(-1)-z(-1)=-328(-328)

(Pretend that the lines are not broken and look like this [])

(⎡ 1 1 1 690 ⎤)

rref (⎢20 15 10 10,030 ⎥)

(⎣ 1 -1 -1 -328 ⎦)

Answer(Matrix): ⎡1 0 0 181 ⎤

⎢0 1 0 264⎥

⎣0 0 1 245⎦

Or you could solve without using a matrix: