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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!

Write the slope-intercept form of the equation of the line through the given points: (0, 1) and (-4, -4); (0, -4) and (5, 2); (1, 5) and (4, 2); (-4, 5) and (-5, -4).

Learn how to write the slope-intercept form of a line passing through points such as (0, 1) and (-4, -4), (0, -4) and (5, 2). This problem involves key concepts like the slope formula and finding the y-intercept, suitable for students in Grades 7-9.

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