A tailor charges a basic fee of $11 plus $3.00 per letter to sew an athlete's name on the back of a jacket.


(a) Write a linear equation that will find the cost *y* to have a name containing *x* letters sewn on the back of a jacket.


y=3x+11
 


(b) Graph the equation.

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(c) Suppose the tailor raises the basic fee to $30. Draw the new graph showing the increased cost.

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### (a) Writing the Linear Equation

The cost to have a name sewn on a jacket is composed of a basic fee plus a fee per letter. The cost function is given as:

- **Basic fee**: $11
- **Cost per letter**: $3 per letter.

Thus, the linear equation that describes the total cost $ y $ as a function of the number of letters $ x $ is:

\[
y = 3x + 11
\]

Where:
- $ y $ is the total cost,
- $ x $ is the number of letters in the name.

### (b) Graphing the Equation

The equation $ y = 3x + 11 $ represents a line with:
- A slope of 3, indicating that for every additional letter, the cost increases by $3.
- A y-intercept of 11, which is the basic fee.

To graph the equation:
1. Start at the point $ (0, 11) $, because when there are 0 letters, the cost is $11.
2. For every increase in $ x $ (number of letters), increase $ y $ by 3 units.

You can plot points such as:
- $ (0, 11) $,
- $ (1, 14) $,
- $ (2, 17) $,
- $ (3, 20) $,
and then draw a straight line through these points.

### (c) New Graph with Basic Fee Raised to $30

If the basic fee is raised to $30, the new equation becomes:

\[
y = 3x + 30
\]

This new line will have:
- The same slope of 3 (cost per letter is unchanged).
- A new y-intercept at 30, which reflects the higher basic fee.

To graph this equation:
1. Start at the point $ (0, 30) $, because when there are 0 letters, the cost is $30.
2. Plot points like:
   - $ (0, 30) $,
   - $ (1, 33) $,
   - $ (2, 36) $,
   - $ (3, 39) $,
and draw the line through these points.

This line will be parallel to the original line but shifted upwards by 19 units (from $11 to $30).

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Do you want help creating the graph? Let me know if you have any questions!

### Related Questions:
1. How would the equation change if the cost per letter increased to $5?
2. What is the slope of the line, and how is it related to the problem?
3. How does the y-intercept affect the cost when no letters are added?
4. Can the equation ever give a negative value for cost? Why or why not?
5. How would you interpret the slope of the line in real-world terms?

**Tip:** Always check the y-intercept to understand the base cost or starting point of a problem like this.

A tailor charges a basic fee of $11 plus $3.00 per letter to sew an athlete's name on the back of a jacket. (a) Write a linear equation that will find the cost y to have a name containing x letters sewn on the back of a jacket. (b) Graph the equation. (c) Suppose the tailor raises the basic fee to $30. Draw the new graph showing the increased cost.

Learn how to model and graph a real-world scenario involving a tailor's pricing structure. The linear equation y = 3x + 11 represents the total cost based on the number of letters in a name. This guide also explores the impact of increasing the basic fee to $30 and how the graph shifts.

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