The total cost, y, for x passes to Astroworld includes a flat fee for processing and a cost per pass. One family purchased 7 passes for a total cost of $500. Another family purchased 10 passes for $620.

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

Group of answer choices

y = 40x + 220

y = 220x+ 40

y=40x

y = 220x

We are given that the total cost $ y $ for $ x $ passes includes a flat fee and a cost per pass, and we need to find the linear equation that represents this cost.

### Step 1: Form a System of Equations
From the problem, we have two situations:
1. A family purchased 7 passes for a total cost of $500.
   This gives the equation:
   \[
   500 = 7m + b
   \]
   where $ m $ is the cost per pass, and $ b $ is the flat fee.

2. Another family purchased 10 passes for a total cost of $620.
   This gives the equation:
   \[
   620 = 10m + b
   \]

### Step 2: Solve the System of Equations
We now have the following system of equations:
\[
500 = 7m + b
\]
\[
620 = 10m + b
\]

Subtract the first equation from the second:
\[
(620 - 500) = (10m + b) - (7m + b)
\]
\[
120 = 3m
\]
Solve for $ m $:
\[
m = \frac{120}{3} = 40
\]

### Step 3: Find the Flat Fee
Substitute $ m = 40 $ into one of the original equations, say $ 500 = 7m + b $:
\[
500 = 7(40) + b
\]
\[
500 = 280 + b
\]
\[
b = 500 - 280 = 220
\]

### Step 4: Write the Equation
Now that we know the cost per pass $ m = 40 $ and the flat fee $ b = 220 $, the equation for the total cost $ y $ in terms of $ x $ passes is:
\[
y = 40x + 220
\]

### Answer:
The correct equation is:
\[
y = 40x + 220
\]

Would you like more details or have any further questions?

### Follow-up Questions:
1. How would the equation change if the flat fee increased?
2. What is the total cost for 15 passes using this equation?
3. If a family only paid the flat fee, what would their cost be?
4. What happens to the cost per pass if the flat fee is reduced to $180?
5. How would the system of equations change if two different families bought different numbers of passes?

### Tip:
When solving a system of linear equations, eliminating one variable by subtraction or substitution can simplify finding both variables.

The total cost, y, for x passes to Astroworld includes a flat fee for processing and a cost per pass. One family purchased 7 passes for a total cost of $500. Another family purchased 10 passes for $620. Which linear equation represents the total cost in dollars, y, for x passes?

Discover how to find the linear equation for the total cost of Astroworld passes using two different scenarios. This problem involves solving a system of equations to determine the cost per pass and the flat fee. Suitable for students in Grades 8-10.

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