The Golden Dome Theatre has a seating capacity of 300 seats.  Adult tickets sell for $5.00 each, and children’s tickets sell for $3.00 each.  The theatre must have sales of at least $1,000 to breakeven for the night.  Write these constraints of this system if x represents the number of adult tickets, and y represents the number of children’s tickets.

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S=5x+3y is revenue for ticket sales. So 5x+3y=1000 and x+y=300 for all seats filled. If all seats are filled we can write 3x+3y=900 and 3x+1000-5x=900, 100=5x-3x, 2x=100, x=50, so y=250. The revenue from adult ticket sales is $250 and from children's tickets is $750. If x=51 and y=249 the total revenue is 255+747=$1002; if x=49 and y=251, total sales=245+753=$998. If adults fill all the seats, the total sales=$1,500 (maximum); if children fill all the seats total sales=$900. So the break-even point is 50 adults and 250 children. If the number of adults is greater than 50, the sales will exceed $1,000 up to a maximum of $1,500 when only adults occupy the seats and all seats are occupied.

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