Let's analyze the questions step-by-step based on the image.

Given Table:

Part A: Finding the y-intercept

The y-intercept is the y-value when $x = 0$. From the table, when $x = 0$, $y = -5.0$.

• Answer for Part A: The y-intercept is -5.0.

Part B: Calculating the Slope

The slope (m) of a linear relationship can be found using the formula: $m = \frac{{y_2 - y_1}}{{x_2 - x_1}}$ Using two points from the table, such as $(-2, -10.0)$ and $(0, -5.0)$: $m = \frac{-5.0 - (-10.0)}{0 - (-2)} = \frac{5.0}{2} = 2.5$

• Answer for Part B: The slope of the line is 2.5.

Part C: Writing the Equation of the Line

Using the slope-intercept form $y = mx + b$, where $m = 2.5$ and $b = -5.0$, the equation becomes: $y = 2.5x - 5.0$

• Answer for Part C: The equation of the line is $y = 2.5x - 5.0$.

Part D: Verifying a Point on the Line

The student claims that the point $(9, -17.5)$ lies on the line. To verify, substitute $x = 9$ into the equation: $y = 2.5 \times 9 - 5.0 = 22.5 - 5.0 = 17.5$ The calculated y-value is $17.5$, not $-17.5$, so the point $(9, -17.5)$ does not lie on the line.

• Answer for Part D: The student is incorrect.

Would you like further details on any part?

Here are some related questions for deeper understanding:

1. How do you determine the y-intercept from a table?
2. Can you find the slope using any two points on a linear line?
3. Why is it essential to use the slope-intercept form to write the equation of a line?
4. How can you verify if a point lies on a line using the line’s equation?
5. What happens to the line if the slope or y-intercept changes?

Tip: When verifying if a point lies on a line, substitute both x and y into the equation to check for equality.