If all 500 seats are filled for a performance, how many of each type of ticket must have been sold for the members to raise exactly $2,050?
Understand the Problem
The question involves setting up a system of two linear equations to determine the number of adult and student tickets that need to be sold to meet specific revenue and capacity constraints. Given that the auditorium capacity is 500 and the revenue goal is $2,050, the problem is asking for us to find how many adult tickets ($6.50 each) and student tickets ($3.50 each) must be sold.
Answer
100 adult tickets and 400 student tickets.
Answer for screen readers
100 adult tickets and 400 student tickets must be sold.
Steps to Solve
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Set up the system of equations Let $d$ be the number of adult tickets and $s$ be the number of student tickets. We know that the total number of tickets sold is 500, so $d + s = 500$. We also know that the total revenue is $2050, so $6.50d + 3.50s = 2050$. Our system of equations is: $$ d + s = 500 $$ $$ 6.50d + 3.50s = 2050 $$
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Solve for one variable in the first equation Solve the first equation for $d$: $$d = 500 - s$$
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Substitute into the second equation Substitute the expression for $d$ into the second equation: $$6.50(500 - s) + 3.50s = 2050$$
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Simplify and solve for s Expand and simplify the equation: $$3250 - 6.50s + 3.50s = 2050$$ $$3250 - 3s = 2050$$ $$-3s = 2050 - 3250$$ $$-3s = -1200$$ $$s = \frac{-1200}{-3}$$ $$s = 400$$
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Solve for d Substitute the value of $s$ back into the equation $d = 500 - s$: $$d = 500 - 400$$ $$d = 100$$
100 adult tickets and 400 student tickets must be sold.
More Information
To verify the solution, we can plug the values of $d$ and $s$ back into the original equations: $100 + 400 = 500$ (Total tickets) $6.50(100) + 3.50(400) = 650 + 1400 = 2050$ (Total revenue)
Tips
A common mistake is to mix up the coefficients in the revenue equation or to make errors in the algebraic manipulation when solving for the variables. Always double-check your work, especially when distributing and combining like terms.
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