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The equations involve finding the linear equations using the **point-slope form**, which is given by:

\[
y - y_1 = m(x - x_1)
\]

where \(m\) is the slope, and \((x_1, y_1)\) is a point on the line.

Let's solve each part one by one:

### 1. \( M = 7 \), point \( (1, -1) \)

Using the formula:
\[
y - (-1) = 7(x - 1)
\]
\[
y + 1 = 7(x - 1)
\]
Expanding:
\[
y + 1 = 7x - 7
\]
\[
y = 7x - 8
\]

The equation of the line is:
\[
y = 7x - 8
\]

---

### 2. \( M = \frac{1}{5} \), point \( (1, 4) \)

Using the point-slope form:
\[
y - 4 = \frac{1}{5}(x - 1)
\]
Expanding:
\[
y - 4 = \frac{1}{5}x - \frac{1}{5}
\]
\[
y = \frac{1}{5}x - \frac{1}{5} + 4
\]
\[
y = \frac{1}{5}x + \frac{19}{5}
\]

The equation of the line is:
\[
y = \frac{1}{5}x + \frac{19}{5}
\]

---

### 3. Two points: \( (2, 3) \) and \( (-1, 0) \)

First, calculate the slope \( M \):
\[
M = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{-1 - 2} = \frac{-3}{-3} = 1
\]

Now, using the point \( (2, 3) \):
\[
y - 3 = 1(x - 2)
\]
\[
y - 3 = x - 2
\]
\[
y = x + 1
\]

The equation of the line is:
\[
y = x + 1
\]

---

### 4. Two points: \( (-3, -1) \) and \( (1, 5) \)

First, calculate the slope \( M \):
\[
M = \frac{5 - (-1)}{1 - (-3)} = \frac{5 + 1}{1 + 3} = \frac{6}{4} = \frac{3}{2}
\]

Now, using the point \( (-3, -1) \):
\[
y - (-1) = \frac{3}{2}(x - (-3))
\]
\[
y + 1 = \frac{3}{2}(x + 3)
\]
Expanding:
\[
y + 1 = \frac{3}{2}x + \frac{9}{2}
\]
\[
y = \frac{3}{2}x + \frac{9}{2} - 1
\]
\[
y = \frac{3}{2}x + \frac{7}{2}
\]

The equation of the line is:
\[
y = \frac{3}{2}x + \frac{7}{2}
\]

---

Would you like any more details or have any questions on these solutions?

Here are 5 related questions to expand this topic:
1. How do you derive the point-slope form from the slope formula?
2. What is the difference between point-slope and slope-intercept forms?
3. How can you convert point-slope form into standard form?
4. What happens when two points have the same x-value (vertical line)?
5. How can you graph a line given in point-slope form?

**Tip:** To verify your work, substitute one of the points back into the final equation to ensure it satisfies the line equation.

Learn how to solve linear equations using the point-slope form formula: y - y1 = M(x - x1). Includes problems involving slopes 7, 1/5, and points (1, -1), (1, 4), as well as solutions for two-point equations like (2, 3) and (-1, 0). Suitable for Grades 8-10.

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