Let's work through the problems step by step. We are asked to find the slope-intercept form of the equation of the line passing through two points for each question. The slope-intercept form is given by:

$y = mx + b$ where $m$ is the slope and $b$ is the y-intercept. We can find $m$ using the formula:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Problem 9: Through (0, 1) and (-4, -4)

• Points: $(0, 1)$ and $(-4, -4)$
• Slope $m$:

$m = \frac{-4 - 1}{-4 - 0} = \frac{-5}{-4} = \frac{5}{4}$

Now, substitute $m = \frac{5}{4}$ and the point $(0, 1)$ into the slope-intercept form:

$y = \frac{5}{4}x + 1$

Problem 10: Through (0, -4) and (5, 2)

• Points: $(0, -4)$ and $(5, 2)$
• Slope $m$:

$m = \frac{2 - (-4)}{5 - 0} = \frac{6}{5}$

Now, substitute $m = \frac{6}{5}$ and the point $(0, -4)$ into the slope-intercept form:

$y = \frac{6}{5}x - 4$

Problem 11: Through (1, 5) and (4, 2)

• Points: $(1, 5)$ and $(4, 2)$
• Slope $m$:

$m = \frac{2 - 5}{4 - 1} = \frac{-3}{3} = -1$

Now, substitute $m = -1$ and the point $(1, 5)$ into the slope-intercept form. First, solve for $b$:

$5 = -1(1) + b \implies b = 6$

The equation is:

$y = -x + 6$

Problem 12: Through (-4, 5) and (-5, -4)

• Points: $(-4, 5)$ and $(-5, -4)$
• Slope $m$:

$m = \frac{-4 - 5}{-5 - (-4)} = \frac{-9}{-1} = 9$

Now, substitute $m = 9$ and the point $(-4, 5)$ into the slope-intercept form. First, solve for $b$:

$5 = 9(-4) + b \implies 5 = -36 + b \implies b = 41$

The equation is:

$y = 9x + 41$

Would you like any further clarifications or details on any of these solutions?

Here are 5 related questions to expand on this:

1. How do you find the slope of a line given two points in general?
2. What are other forms of linear equations besides slope-intercept form?
3. Can you explain how to find the y-intercept from any two points on a line?
4. How does changing the slope affect the steepness of the line?
5. What is the relationship between parallel lines and their slopes?

Tip: When using the slope formula, be careful with the signs of the coordinates to avoid calculation errors!