Question 145883: For a recent major league baseball game, a total of 52,000 tickets were sold. Tickets were sold in three categories: Box seats for $90 each, Bleacher seats for $40 each, and Balcony seats for $60 each. A total of $3,320,000 was collected from the sale of the tickets. The combined number of Box and Balcony seats exceeded twice the number of Bleacher seats by 10,000. Find the number of each of the three types of tickets sold.
Answer by nerdybill(7384)   (Show Source):
You can put this solution on YOUR website! For a recent major league baseball game, a total of 52,000 tickets were sold. Tickets were sold in three categories: Box seats for $90 each, Bleacher seats for $40 each, and Balcony seats for $60 each. A total of $3,320,000 was collected from the sale of the tickets. The combined number of Box and Balcony seats exceeded twice the number of Bleacher seats by 10,000. Find the number of each of the three types of tickets sold.
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Looks like you have 3 variables, so you'll need 3 equations.
And, it appears the best way to solve this is with matrices.
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Let x = number of box seats sold
and y = number of bleacher seats sold
and z = number of balcony seats sold
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from:"a total of 52,000 tickets were sold." we get:
x + y + z = 52000
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from:cost of each ticket and total $ collected we get:
90x + 40y + 60z = 3320000
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from:"combined number of Box and Balcony seats exceeded twice the number of Bleacher seats by 10,000"
x + z = 2y+10000
x - 2y + z = 10000
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Our coefficient matrix is:
1 1 1
90 40 60
1 -2 1
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determinant of the coef matrix:
(40+120)-(90-60)+(-180-40)
=160-(30)+(-220)
=160-30-220
=160-250
=-90
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Our matrix for "box seats" (x) is:
52000 1 1
3320000 40 60
10000 -2 1
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determinant of box seats:
52000[40+120] - [3320000 - 600000] + [-6640000 - 400000]
52000[160] - [2720000] + [-7040000]
-1440000
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box seats sold = "det of box seats"/"det of coef"
box seats sold = -1440000/-90
box seats sold = 16000
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I'll let you finish it from here. It is similar to the above:
bleacher seats sold = "det of bleacher seats"/"det of coef"
balcony seats sold = "det of balcony seats"/"det of coef"
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