Question

A theater is selling tickets for a performance. Mr. Smith purchased 8 senior tickets and 5 child tickets for 136$ for his friends and family. Mr. Jackson purchased 4 senior tickets and 6 child tickets for 96$. What is the price of a senior ticket?

Answer 1

Let \( x \) be the price of a senior ticket and \( y \) be the price of a child ticket. We can set up the following system of equations based on the information provided:

1. From Mr. Smith's purchase:
\[
8x + 5y = 136
\]

2. From Mr. Jackson's purchase:
\[
4x + 6y = 96
\]

Now we can solve this system of equations.

First, we can simplify the second equation. Let's divide the entire equation by 2:
\[
2x + 3y = 48 \quad \text{(Equation 2)}
\]

Now we have the system:
1. \( 8x + 5y = 136 \) (Equation 1)
2. \( 2x + 3y = 48 \) (Equation 2)

Next, we'll solve Equation 2 for \( y \):
\[
3y = 48 - 2x
\]
\[
y = \frac{48 - 2x}{3}
\]

Now, substitute this expression for \( y \) into Equation 1:
\[
8x + 5\left(\frac{48 - 2x}{3}\right) = 136
\]

Multiply everything by 3 to eliminate the fraction:
\[
24x + 5(48 - 2x) = 408
\]
\[
24x + 240 - 10x = 408
\]
\[
14x + 240 = 408
\]

Now, isolate \( x \):
\[
14x = 408 - 240
\]
\[
14x = 168
\]
\[
x = \frac{168}{14} = 12
\]

Now that we have the price of a senior ticket \( (x) \), we can find the price of a child ticket \( (y) \) by substituting \( x = 12 \) back into Equation 2:
\[
2(12) + 3y = 48
\]
\[
24 + 3y = 48
\]
\[
3y = 48 - 24
\]
\[
3y = 24
\]
\[
y = 8
\]

Thus, the price of a senior ticket is \( \boxed{12} \) dollars.