The total cost, y, for x passes to the State Fair includes a flat fee for processing and a cost per pass. One family purchased 5 passes for a total cost of $390. Another family purchased 8 passes for $615.

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

We are given the following information:

- A family bought 5 passes for a total cost of $390.
- Another family bought 8 passes for a total cost of $615.

The total cost, $ y $, includes a **flat processing fee** and a **cost per pass**. This implies that the total cost can be modeled by the linear equation:

\[
y = mx + b
\]

Where:
- $ y $ is the total cost,
- $ x $ is the number of passes,
- $ m $ is the cost per pass, and
- $ b $ is the flat processing fee.

### Step 1: Set up the system of equations
Using the information given, we can form two equations based on the two families’ purchases:

1. For 5 passes ($x = 5$) costing $390 ($y = 390$):
\[
390 = 5m + b
\]

2. For 8 passes ($x = 8$) costing $615 ($y = 615$):
\[
615 = 8m + b
\]

### Step 2: Solve the system of equations
We will solve these two equations to find $m$ (the cost per pass) and $b$ (the flat processing fee).

1. First equation: 
\[
390 = 5m + b \quad \text{(Equation 1)}
\]

2. Second equation:
\[
615 = 8m + b \quad \text{(Equation 2)}
\]

We can subtract Equation 1 from Equation 2 to eliminate $b$:

\[
(615 - 390) = (8m - 5m)
\]

\[
225 = 3m
\]

\[
m = \frac{225}{3} = 75
\]

So, the cost per pass $ m = 75 $.

### Step 3: Find the flat processing fee $b$
Now, substitute $m = 75$ into one of the original equations. Using Equation 1:

\[
390 = 5(75) + b
\]

\[
390 = 375 + b
\]

\[
b = 390 - 375 = 15
\]

So, the flat processing fee $ b = 15 $.

### Final Linear Equation
The linear equation representing the total cost $ y $ for $ x $ passes is:

\[
y = 75x + 15
\]

This equation shows that the total cost is $75 per pass with a $15 flat processing fee.

---

Would you like more details, or do you have any questions?  
Here are 5 related questions to expand your understanding:

1. How would the total cost change if the flat fee increased?
2. What would the total cost be for 10 passes?
3. How can you graph this equation to show the relationship between cost and number of passes?
4. What would happen if there was no flat fee and only a cost per pass?
5. How do systems of linear equations apply in other real-world scenarios?

**Tip:** When solving a system of linear equations, elimination or substitution methods help simplify and find solutions efficiently.

The total cost, y, for x passes to the State Fair includes a flat fee for processing and a cost per pass. One family purchased 5 passes for a total cost of $390. Another family purchased 8 passes for $615. Which linear equation represents the total cost in dollars, y, for x passes?

Learn how to solve a linear equation representing the total cost of State Fair passes, using real-life examples of two families purchasing 5 and 8 passes. This problem involves key algebraic concepts, including linear equations and systems of equations. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation