Mrs. Gallas has custom t-shirts made each year for her AP Stats class. She uses a local screen-printing company to make the shirts. They charge a fee to set up the printer and \$6.50 $ 6.50 $6.50per shirt. Last year, she ordered 89 89 89shirts which cost \$608.49 $ 608.49 $608.49. a. Write an equation in point-slope form for the total cost, y 𝑦 y, for x 𝑥 xshirts. b. How much is the fee to set up the printer? c. Write an equation in slope-intercept form for the total cost, y 𝑦 y, for x 𝑥 xshirts.

Let's go step by step through the problem:

### Part (a): Write an equation in point-slope form for the total cost $ y $ for $ x $ shirts.

The **point-slope form** of a linear equation is:

\[
y - y_1 = m(x - x_1)
\]

Where:
- $ m $ is the slope (which represents the cost per shirt),
- $ (x_1, y_1) $ is a point on the line (representing the number of shirts ordered and the total cost).

We are given that the cost per shirt is \$6.50, so $ m = 6.50 $. Also, we know that for $ x_1 = 89 $ shirts, the total cost $ y_1 = 608.49 $.

Thus, the equation in point-slope form becomes:

\[
y - 608.49 = 6.50(x - 89)
\]

### Part (b): How much is the fee to set up the printer?

The **setup fee** is a fixed cost that doesn't depend on the number of shirts ordered. We can calculate it by using the total cost for 89 shirts and subtracting the cost for the shirts themselves.

The total cost is \$608.49 for 89 shirts. The cost for 89 shirts is:

\[
\text{Cost for 89 shirts} = 6.50 \times 89 = 578.50
\]

The setup fee is the difference between the total cost and the cost for the shirts:

\[
\text{Setup fee} = 608.49 - 578.50 = 29.99
\]

So, the setup fee is **\$29.99**.

### Part (c): Write an equation in slope-intercept form for the total cost, $ y $, for $ x $ shirts.

The **slope-intercept form** of a linear equation is:

\[
y = mx + b
\]

Where:
- $ m $ is the slope (the cost per shirt), and
- $ b $ is the y-intercept (the setup fee).

We already know that $ m = 6.50 $ and the setup fee $ b = 29.99 $. So, the equation in slope-intercept form is:

\[
y = 6.50x + 29.99
\]

### Summary:
1. **Point-slope form**: $ y - 608.49 = 6.50(x - 89) $
2. **Setup fee**: \$29.99
3. **Slope-intercept form**: $ y = 6.50x + 29.99 $

Do you have any questions or would you like further details?

---

**Related questions:**
1. What would the total cost be if Mrs. Gallas ordered 100 shirts?
2. How many shirts could Mrs. Gallas order if her budget is \$800?
3. What happens to the total cost equation if the price per shirt increases to \$7.00?
4. How can we graph the relationship between the number of shirts and the total cost?
5. Can you derive the point-slope form from the slope-intercept form?

**Tip**: The slope in a cost function often represents the variable cost per item, while the intercept usually represents fixed costs.

Mrs. Gallas has custom t-shirts made each year for her AP Stats class. She uses a local screen-printing company to make the shirts. They charge a fee to set up the printer and $6.50 per shirt. Last year, she ordered 89 shirts which cost $608.49. a. Write an equation in point-slope form for the total cost, y, for x shirts. b. How much is the fee to set up the printer? c. Write an equation in slope-intercept form for the total cost, y, for x shirts.

Learn how to calculate the total cost for custom t-shirts using point-slope and slope-intercept forms. This problem involves a fixed setup fee and a per-shirt cost, ideal for students in Grades 9-11 exploring linear equations.

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