Question

Tickets for a basketball game cost $20 for upper-level seats and $30 for lower-level seats. Tickets at the same venue for an ice hockey game cost $15 for upper-level seats and $35 for lower-level seats. If tickets for all seats are sold, the venue generates $7000 in revenue for a basketball game and $6500 for a hockey game. How many upper and lower-level seats does the venue have?

Let x represent the number of upper-level seats.

Let y represent the number of lower-level seats.

20x + 30y = 7000
15x + 35y = 6500

The venue has
upper-level seats and lower-level seats.

Answer 1

Answer: 300 seats

Explanation:

Easy!

20x + 30y = 7000

15x + 35y = 6500

Simplify by division of a common factor:

5(4x + 6y = 1400)

5(3x + 7x = 1300)

Choose an equation to solve for a variable and use substitution:

4x = 1400 - 6y

x = 350 - 3/2y

3(350 - 3/2y) + 7y = 1300

1050 - 9/2y + 7y = 1300

5/2y = 250

y = 100

Now substitute the solution in both equations to see if it works:

20x + 30(10) = 7000

20x + 3000 = 7000

20x = 4000

2x = 400

x = 200

Now for the other equation:

15x + 35(100) = 6500

15x + 3500 = 6500

15x = 3000

x = 200

Now add x to y, as the total number of seats is equal to x + y.

x + y = total number of seats.

200 + 100 = 300

Therefore, there must be 300 seats in total.

Answer 2

x = 200

y = 100

Explanation:

20x + 30y = 7000 Divide by 10

15x + 35y = 6500 Divide by 5

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It helps if you break down both equations to something simpler. It looks like not always.

(2x + 3y = 700) Multiply by 3

(3x + 7y = 1300) Multiply by 2

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6x + 9y = 2100

6x + 14y =2600 Subtract

-5y = - 500 Divide by -5

-5y/-5 = -500/-5

y = 100

2x + 3y = 700 substitute y = 100

2x+ 3*100 = 700 Combine

2x + 300 = 700 Subtract 300 from both sides.

2x = 700 - 300

2x = 400

x = 400/2

x = 200

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Hockey check

15x + 35y = 6500

15*200 + 35*100 = 6500

3000 + 3500 = 6500 Just as it should.