Question

The total cost, y, for x tickets to a concert includes a flat fee for processing and a cost per ticket. One customer purchased 4 tickets for a total cost of $160. Another customer purchased 8 tickets for a total cost of $300.

Which linear equation represents the total cost in dollars, y, for x tickets?

Question

Answer 1

To find the linear equation that represents the total cost \( y \) for \( x \) tickets, we can express the total cost as:

\[
y = mx + b
\]

where:
- \( m \) is the cost per ticket,
- \( b \) is the flat fee for processing.

We have two pieces of data from the problem:

1. For 4 tickets, the total cost is $160. This gives us the equation:
\[
160 = 4m + b \quad (1)
\]

2. For 8 tickets, the total cost is $300. This gives us the equation:
\[
300 = 8m + b \quad (2)
\]

Now we have a system of two equations. We can solve these equations to find \( m \) and \( b \).

First, let's subtract equation (1) from equation (2):

\[
300 - 160 = (8m + b) - (4m + b)
\]
\[
140 = 4m
\]

Now, solving for \( m \):

\[
m = \frac{140}{4} = 35
\]

Now that we have \( m = 35 \) (the cost per ticket), we can substitute \( m \) back into one of the original equations to find \( b \). We'll use equation (1):

\[
160 = 4(35) + b
\]
\[
160 = 140 + b
\]
\[
b = 160 - 140 = 20
\]

Now we have \( b = 20 \) (the flat fee for processing) and \( m = 35 \).

Thus, the linear equation that represents the total cost in dollars \( y \) for \( x \) tickets is:

\[
y = 35x + 20
\]