Let's solve the problem shown in the image step-by-step.

Problem:

Find the equation of the line passing through the points (-3, 7) and (2, -6).

1. Find the slope of the line: The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is: $m = \frac{y_2 - y_1}{x_2 - x_1}$

   Substituting the points $(-3, 7)$ and $(2, -6)$: $m = \frac{-6 - 7}{2 - (-3)} = \frac{-6 - 7}{2 + 3} = \frac{-13}{5}$ So, the slope $m = \frac{-13}{5}$.

2. Point-slope form: The point-slope form of a line is given by: $y - y_1 = m(x - x_1)$ Using the point $(-3, 7)$ and the slope $m = \frac{-13}{5}$: $y - 7 = \frac{-13}{5}(x - (-3)) = \frac{-13}{5}(x + 3)$ This is the equation in point-slope form.

3. Slope-intercept form: To convert the equation to slope-intercept form ($y = mx + b$), we can simplify: $y - 7 = \frac{-13}{5}(x + 3)$ Expanding the right side: $y - 7 = \frac{-13}{5}x - \frac{39}{5}$ Adding 7 to both sides (which is equivalent to $\frac{35}{5}$): $y = \frac{-13}{5}x - \frac{39}{5} + \frac{35}{5} = \frac{-13}{5}x - \frac{4}{5}$ So, the slope-intercept form is: $y = \frac{-13}{5}x - \frac{4}{5}$

Summary of the answers:

• (a) The slope is $\frac{-13}{5}$.
• (b) The point-slope form is $y - 7 = \frac{-13}{5}(x + 3)$.
• (c) The slope-intercept form is $y = \frac{-13}{5}x - \frac{4}{5}$.

Would you like further clarification or details on any part?

Follow-up questions:

1. How is the point-slope form of a line derived?
2. What are the key differences between point-slope form and slope-intercept form?
3. Can a line passing through two points ever have an undefined slope?
4. How can you verify the equation of a line given two points?
5. What happens if two points have the same $x$-coordinate?

Tip:

Always double-check your calculations for the slope, as a small mistake can change the entire equation of the line!