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Let's solve each problem step by step using the point-slope form of a line:

🔹 Point-Slope Form:

The point-slope form of a line is:
$$
y - y_1 = m(x - x_1)
$$
where:
- $ m $ is the slope,
- $ (x_1, y_1) $ is a point on the line.

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✔ Part 1: Write an equation in point-slope form

#### 1. Passes through (2, 3) and has slope of $-\frac{1}{2}$

Use $ m = -\frac{1}{2} $, $ (x_1, y_1) = (2, 3) $

$$
y - 3 = -\frac{1}{2}(x - 2)
$$

✔ Answer: $ y - 3 = -\frac{1}{2}(x - 2) $

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#### 2. Passes through (-1, 4) and $ m = 4 $

$$
y - 4 = 4(x + 1)
$$

✔ Answer: $ y - 4 = 4(x + 1) $

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#### 3. Passes through (0, 2) and has slope of $-\frac{5}{3}$

$$
y - 2 = -\frac{5}{3}(x - 0)
$$

Simplify:
$$
y - 2 = -\frac{5}{3}x
$$

✔ Answer: $ y - 2 = -\frac{5}{3}x $

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#### 4. Passes through (4, -2) and $ m = 0 $

A slope of 0 means a horizontal line.

$$
y + 2 = 0(x - 4)
$$

$$
y + 2 = 0 \quad \Rightarrow \quad y = -2
$$

But we want point-slope form, so:

$$
y + 2 = 0(x - 4)
$$

✔ Answer: $ y + 2 = 0(x - 4) $

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#### 5. Passes through (-4, 6) and (-2, 5)

First, find the slope $ m $:

$$
m = \frac{5 - 6}{-2 - (-4)} = \frac{-1}{2} = -\frac{1}{2}
$$

Now use one point, say $(-4, 6)$:

$$
y - 6 = -\frac{1}{2}(x + 4)
$$

✔ Answer: $ y - 6 = -\frac{1}{2}(x + 4) $

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#### 6. Passes through (-1, -7) and (1, 3)

Find slope:

$$
m = \frac{3 - (-7)}{1 - (-1)} = \frac{10}{2} = 5
$$

Use point $(-1, -7)$:

$$
y + 7 = 5(x + 1)
$$

✔ Answer: $ y + 7 = 5(x + 1) $

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✔ Part 2: Give the slope and a point on each line

We are given equations in point-slope form. Recall:
$$
y - y_1 = m(x - x_1)
$$
So:
- The number multiplied by $ (x - x_1) $ is the slope $ m $.
- The point is $ (x_1, y_1) $, found from $ (x - x_1) $ and $ y - y_1 $.

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#### 7. $ y + 2 = \frac{2}{3}(x - 4) $

Rewrite as: $ y - (-2) = \frac{2}{3}(x - 4) $

- Slope $ m = \frac{2}{3} $
- Point: $ (4, -2) $

✔ Slope: $ \frac{2}{3} $, Point: $ (4, -2) $

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#### 8. $ y - 3 = \frac{1}{2}(x - 3) $

- Slope $ m = \frac{1}{2} $
- Point: $ (3, 3) $

✔ Slope: $ \frac{1}{2} $, Point: $ (3, 3) $

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#### 9. $ y + 5 = \frac{1}{4}(x + 2) $

Rewrite: $ y - (-5) = \frac{1}{4}(x - (-2)) $

- Slope $ m = \frac{1}{4} $
- Point: $ (-2, -5) $

✔ Slope: $ \frac{1}{4} $, Point: $ (-2, -5) $

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#### 10. $ y = 2(x + 3) $

Rewrite: $ y - 0 = 2(x + 3) $ → $ y - 0 = 2(x - (-3)) $

- Slope $ m = 2 $
- Point: $ (-3, 0) $

✔ Slope: $ 2 $, Point: $ (-3, 0) $

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#### 11. $ y - 8 = -3(x + 1) $

Rewrite: $ y - 8 = -3(x - (-1)) $

- Slope $ m = -3 $
- Point: $ (-1, 8) $

✔ Slope: $ -3 $, Point: $ (-1, 8) $

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#### 12. $ y + 3 = -\frac{1}{5}x $

Rewrite: $ y - (-3) = -\frac{1}{5}(x - 0) $

- Slope $ m = -\frac{1}{5} $
- Point: $ (0, -3) $

✔ Slope: $ -\frac{1}{5} $, Point: $ (0, -3) $

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✔ Final Answers Summary

#### Part 1: Equations in Point-Slope Form

1. $ y - 3 = -\frac{1}{2}(x - 2) $
2. $ y - 4 = 4(x + 1) $
3. $ y - 2 = -\frac{5}{3}x $
4. $ y + 2 = 0(x - 4) $
5. $ y - 6 = -\frac{1}{2}(x + 4) $
6. $ y + 7 = 5(x + 1) $

#### Part 2: Slope and Point

7. Slope: $ \frac{2}{3} $, Point: $ (4, -2) $
8. Slope: $ \frac{1}{2} $, Point: $ (3, 3) $
9. Slope: $ \frac{1}{4} $, Point: $ (-2, -5) $
10. Slope: $ 2 $, Point: $ (-3, 0) $
11. Slope: $ -3 $, Point: $ (-1, 8) $
12. Slope: $ -\frac{1}{5} $, Point: $ (0, -3) $

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Let me know if you'd like these converted to slope-intercept form too!