Write a system of equations to describe the situation below, solve using substitution, and fill in the blanks.

Kamal plans to attend the Johnson County Fair and is trying to decide what would be a better deal. He can pay $40 for unlimited rides, or he can pay $12 for admission plus $1 per ride. If Kamal goes on a certain number of rides, the two options wind up costing him the same amount. What is that cost? How many rides is that?

To solve this problem, we will set up a system of equations to represent the costs of the two options and then use substitution to find the number of rides that make the costs equal.

### Step 1: Define the Variables

Let's define:
- $ r $ as the number of rides Kamal takes.
- $ C $ as the total cost.

### Step 2: Set Up the Equations

We have two different pricing options:

1. **Unlimited Rides Option**: This costs $40 regardless of the number of rides.
   \[
   C = 40
   \]

2. **Pay-Per-Ride Option**: This costs $12 for admission plus $1 per ride.
   \[
   C = 12 + 1 \cdot r = 12 + r
   \]

### Step 3: Form the System of Equations

To find the number of rides ($ r $) for which the costs are the same, we set the two equations equal to each other:

\[
40 = 12 + r
\]

### Step 4: Solve for $ r $ Using Substitution

Solve for $ r $:

\[
40 = 12 + r
\]

Subtract 12 from both sides:

\[
40 - 12 = r
\]

\[
28 = r
\]

So, Kamal needs to go on **28 rides** for the costs to be equal.

### Step 5: Find the Cost at This Number of Rides

To find the cost for 28 rides, substitute $ r = 28 $ into either cost equation (since at 28 rides, both costs are equal):

\[
C = 40
\]

Thus, the cost is **$40**.

### Summary

- **Number of rides**: 28
- **Cost**: $40

### Solution to Fill in the Blanks:

1. **Number of rides**: 28 rides
2. **Cost**: $40

Would you like more details, or do you have any questions?

### Related Questions:

1. What would be the cost if Kamal decides to go on 15 rides?
2. If the admission fee for the pay-per-ride option increases to $15, how many rides would make both options equal?
3. How much would Kamal save by choosing the unlimited rides option if he plans to go on 35 rides?
4. If the price per ride in the pay-per-ride option is reduced to $0.50, what would be the new break-even number of rides?
5. How would the equation change if there was a discount on the unlimited rides option?

### Tip:

When solving systems of equations, substitution is a powerful method especially when one of the equations is already solved for a variable.

Learn how to solve a system of linear equations using substitution to determine the break-even point for two pricing options at a fair. This problem demonstrates algebraic concepts and the substitution method, suitable for students in Grades 6-8.

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