https://file.aiquickdraw.com/mathbot/file/17261468292465mszyhc9.jpg

Let's solve this income tax problem step by step based on the information provided in the image.

### Given:
- A taxpayer owes **$1700** plus **18% of the amount of their income over $13,000**.
- You are asked to complete the table for the tax, $ T $, on various incomes, $ I $.

### Step 1: Determine how much income exceeds $13,000
The tax calculation is based on the amount of income over $13,000. For each income $ I $, calculate the excess over $13,000.

For example, if income $ I = 15,000 $:
\[
I - 13,000 = 15,000 - 13,000 = 2,000
\]

### Step 2: Calculate the total tax, $ T $
The total tax is the base tax of $1700 plus 18% of the excess over $13,000.

\[
T = 1700 + 0.18 \times (I - 13,000)
\]

### Step 3: Fill in the table

#### For $ I = 15,000 $:
\[
T = 1700 + 0.18 \times (15,000 - 13,000) = 1700 + 0.18 \times 2,000 = 1700 + 360 = 2060
\]

#### For $ I = 20,000 $:
\[
T = 1700 + 0.18 \times (20,000 - 13,000) = 1700 + 0.18 \times 7,000 = 1700 + 1,260 = 2960
\]

#### For $ I = 22,000 $:
\[
T = 1700 + 0.18 \times (22,000 - 13,000) = 1700 + 0.18 \times 9,000 = 1700 + 1,620 = 3320
\]

Thus, the completed table is:

| $ I $      | $ T $  |
|--------------|----------|
| $ 15,000 $ | $ 2060 $ |
| $ 20,000 $ | $ 2960 $ |
| $ 22,000 $ | $ 3320 $ |

---

### Part b: Write a linear equation in point-slope form
We know that the tax $ T $ is a linear function of income $ I $. Using point-slope form $ T - T_1 = m(I - I_1) $, where the slope $ m $ is 0.18, and $ T_1 = 1700 $ when $ I_1 = 13,000 $.

The linear equation is:
\[
T - 1700 = 0.18(I - 13,000)
\]

### Part c: Convert to slope-intercept form
Solve for $ T $ to convert the equation into slope-intercept form:

\[
T = 0.18I - 0.18 \times 13,000 + 1700
\]
\[
T = 0.18I - 2340 + 1700
\]
\[
T = 0.18I - 640
\]

So, the slope-intercept form of the equation is:
\[
T = 0.18I - 640
\]

---

Would you like further clarifications or more details? Here are 5 related questions:

1. What is the significance of the slope $ 0.18 $ in the tax equation?
2. How can we interpret the y-intercept $ -640 $ in the slope-intercept form?
3. How would the equation change if the base tax increased from $1700 to $2000?
4. How would the tax calculation change for incomes below $13,000?
5. Can you find the income at which the tax becomes exactly $3,000?

**Tip:** Always ensure that point-slope and slope-intercept forms are consistent by plugging values to check accuracy!

A typical income tax schedule is shown, where a single taxpayer owes $1700 plus 18% of the amount of their income over $13,000. Complete the table of values for the tax, T, on various incomes I: $15,000, $20,000, $22,000. Write a linear equation in point-slope form for the amount of tax T a person with an income I would pay if their income is over $13,000. Transform the equation to slope-intercept form.

Solve a linear equation related to income tax where a taxpayer owes $1700 plus 18% of the amount over $13,000. Learn how to complete a tax table and convert from point-slope to slope-intercept form. Includes step-by-step calculations for various incomes such as $15,000, $20,000, and $22,000.

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