Solve System of Equations for Ticket Sales

Math Problem Statement

A school fund raising event sold a total of 212 tickets and generated a total revenue of $1,080.50. There are two types of tickets: adult tickets and child tickets. Each adult ticket costs $6.85, and each child ticket costs $3.90. Write and solve a system of equations to find how many adult and child tickets were sold.

Solution

Let's define variables to represent the number of adult and child tickets sold:

Let $x$ represent the number of adult tickets sold.
Let $y$ represent the number of child tickets sold.

We are given the following information:

The total number of tickets sold is 212, so: $x + y = 212$
The total revenue generated is $1,080.50. Each adult ticket costs $6.85, and each child ticket costs $3.90, so: $6.85x + 3.90y = 1080.50$

We now have a system of two equations:

x + y &= 212 \\ 6.85x + 3.90y &= 1080.50 \end{aligned}$$ ### Solving this system: 1. Solve the first equation for $$y$$: $$y = 212 - x$$ 2. Substitute this expression for $$y$$ into the second equation: $$6.85x + 3.90(212 - x) = 1080.50$$ 3. Expand and solve for $$x$$: $$6.85x + 3.90 \times 212 - 3.90x = 1080.50$$ $$6.85x + 826.80 - 3.90x = 1080.50$$ $$(6.85x - 3.90x) + 826.80 = 1080.50$$ $$2.95x + 826.80 = 1080.50$$ $$2.95x = 1080.50 - 826.80$$ $$2.95x = 253.70$$ $$x = \frac{253.70}{2.95}$$ $$x \approx 86$$ 4. Substitute $$x = 86$$ back into the first equation: $$86 + y = 212$$ $$y = 212 - 86$$ $$y = 126$$ ### Solution: - 86 adult tickets were sold. - 126 child tickets were sold. Would you like further details on the solution or any clarifications? Here are some related questions: 1. What would the revenue be if all tickets were adult tickets? 2. How much more revenue is generated per adult ticket compared to a child ticket? 3. What is the percentage of child tickets sold out of the total? 4. If the ticket prices changed, how would the equations be adjusted? 5. How could you verify this solution using a different method? **Tip:** When solving systems of equations, substitution and elimination methods are useful depending on the situation.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Systems of Equations

Formulas

x + y = total tickets
6.85x + 3.90y = total revenue

Theorems

Substitution Method for Solving Systems of Equations

Suitable Grade Level

Grades 8-10

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