A theater has a seating capacity of 900. They charge $4/ticket for children, $6/

Question

A theater has a seating capacity of 900. They charge $4/ticket for children, $6/ticket for students and $8/ticket for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The ticket receipts totaled $5600. How many children attended the show? Solve this problem by using a system of linear equations. Clearly show how many variables you are using and how you are forming the matrices. 1. Displaythematrices. 2. Display the dimension of the matrices. 3. Displaytheresult.

Explanation / Answer

x = children's tickets
y = students' tickets
z = adults' tickets

x + y + z = 900
4x + 6y + 8z = 5600

z = (1/2)(x + y)

x + y + (1/2)(x + y) = 900 <==> 2x + 2y + x + y = 1800 <==> 3x + 3y = 1800 <==> x + y = 600
4x + 6y + 8(1/2)(x + y) = 5600 <=> 4x + 6y + 4x + 4y = 5600 <=> 8x + 10y = 5600 <=> 4x + 5y = 2800

From the first one y = 600 - x, so let's substitute that into the second one:

4x + 5(600 - x) = 2800
4x + 3000 - 5x = 2800
200 = x ==> this is the number of children's tickets sold

y = 600 - 200 = 400

z = 900 - 400 - 200 = 300

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