Using Linear Systems to Solve Problems

Question

Seats closest to the stage at Dallas's Gexa Energy Pavilion cost $120. The seats in the next section cost $45, and lawn seats are $35. There are twice as many seats in the $45 section as the $120 section. When 20,000 seats are sold, the amphitheater makes $962,500. Determine the number of seats in each of the 3 sections.

Answer 1

Hi Carla,

 

Let there be x seats at the $120 section.

 

There are twice as many $45 seats than $120 seats, so there are 2x $45 seats.

 

Let there be y $35 lawn seats.

 

20,000 seats are sold. That means, x + 2x + y = 20,000

 

The amphitheater made $962,500. So, 120x + 45(2x) + 35y = 962500

 

We can write these equations as

 

3x + y = 20,000

210x + 35y = 962500

 

Let's multiply the top equation by -70,

 

-210x -70y = -1,400,000

210x + 35y = 962500

 

Now add the two equations, noting that the x terms cancel.

 

-35y = -437,500

 

y = 12,500 seats

 

We know earlier that 3x + y = 20,000. So, x = (1/3)(20,000 - y)

x = (1/3)(20,000 - 12,500)

 

x = 2500

 

So, we have 2500 seats sold at $120 each;

2*2500 = 5000 seats sold at $45 each;

12,500 lawn seats sold at $35.