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:

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!