Comprehensive guide to understanding slope, linear equations, and relationships between lines in a coordinate plane.

The image you've uploaded is a comprehensive study guide or reference sheet on linear equations and slope, covering key concepts such as:

- Slope calculation
- Equation of a line in different forms
- Parallel and perpendicular lines

Let’s go through each section, solve the problems, and explain the solutions step-by-step.

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🔹 Top Row: Slope Basics

#### 1. Slope Formula
$$
m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}
$$

Example:
Given points $(1, 2)$ and $(3, 4)$:
$$
m = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1
$$
✔ Slope = 1

#### 2. Slope from Graph
Points: $(-4, 4)$ and $(3, 1)$
$$
m = \frac{1 - 4}{3 - (-4)} = \frac{-3}{7} = -\frac{3}{7}
$$
✔ Slope = $-\frac{3}{7}$

#### 3. Signs of Slope
- Positive slope: line rises from left to right.
- Negative slope: line falls from left to right.
- Zero slope: horizontal line ($y = a$).
- Undefined slope: vertical line ($x = a$).

#### 4. Mr. Slope Mnemonic
A visual tool to remember:
- Positive (+) → up-right
- Negative (−) → down-right
- Zero (0) → flat
- Undefined (U) → vertical

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🔹 Middle Row: Equations of Lines

#### 1. Slope-Intercept Form: $y = mx + b$

> Find an equation of a line with slope -3 passing through (-1, 5).

Use:
$$
y = mx + b
$$
Plug in $m = -3$, $x = -1$, $y = 5$:
$$
5 = -3(-1) + b \Rightarrow 5 = 3 + b \Rightarrow b = 2
$$
So,
$$
\boxed{y = -3x + 2}
$$

✔ Answer: $y = -3x + 2$

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#### 2. Parallel Line Problem

> Find an equation of the line parallel to $y = -\frac{1}{2}x + 2$ passing through (8, -3).

- Parallel lines have same slope.
- So, $m = -\frac{1}{2}$
- Use point-slope form or slope-intercept:

$$
y = mx + b \Rightarrow -3 = -\frac{1}{2}(8) + b \Rightarrow -3 = -4 + b \Rightarrow b = 1
$$

So,
$$
\boxed{y = -\frac{1}{2}x + 1}
$$

✔ Answer: $y = -\frac{1}{2}x + 1$

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#### 3. Parallel Lines Concept

> Two non-vertical lines are parallel if and only if their slopes are equal:
$$
m_1 = m_2
$$
Visual shows two red lines with same slope — they never meet.

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🔹 Bottom Row: Point-Slope Form & Perpendicular Lines

#### 1. Point-Slope Form: $y - y_1 = m(x - x_1)$

> Find an equation of a line through (-2, 7) with slope -5.

Use:
$$
y - y_1 = m(x - x_1)
\Rightarrow y - 7 = -5(x + 2)
$$
Distribute:
$$
y - 7 = -5x - 10 \Rightarrow y = -5x - 3
$$

✔ Answer: $y = -5x - 3$

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#### 2. Perpendicular Line Problem

> Find an equation of a line perpendicular to $3x - 2y = 6$, passing through (1, 2).

Step 1: Rewrite given equation in slope-intercept form.
$$
3x - 2y = 6 \Rightarrow -2y = -3x + 6 \Rightarrow y = \frac{3}{2}x - 3
$$
So, slope of original line: $m_1 = \frac{3}{2}$

Step 2: Perpendicular lines have negative reciprocal slopes.
$$
m_2 = -\frac{2}{3}
$$

Step 3: Use point-slope form with point (1, 2):
$$
y - 2 = -\frac{2}{3}(x - 1)
$$
Distribute:
$$
y - 2 = -\frac{2}{3}x + \frac{2}{3}
\Rightarrow y = -\frac{2}{3}x + \frac{2}{3} + 2
\Rightarrow y = -\frac{2}{3}x + \frac{8}{3}
$$

✔ Answer: $y = -\frac{2}{3}x + \frac{8}{3}$

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#### 3. Perpendicular Lines Concept

> Two non-vertical lines are perpendicular if and only if the product of their slopes is $-1$:
$$
m_1 \cdot m_2 = -1
$$
For example:
- If $m_1 = \frac{3}{2}$, then $m_2 = -\frac{2}{3}$, since $\frac{3}{2} \cdot -\frac{2}{3} = -1$

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✔ Summary of All Answers:

| Problem | Answer |
|--------|--------|
| Slope between (1,2) and (3,4) | $m = 1$ |
| Slope between (-4,4) and (3,1) | $m = -\frac{3}{7}$ |
| Line with slope -3 through (-1,5) | $y = -3x + 2$ |
| Line parallel to $y = -\frac{1}{2}x + 2$ through (8,-3) | $y = -\frac{1}{2}x + 1$ |
| Line through (-2,7) with slope -5 | $y = -5x - 3$ |
| Line perpendicular to $3x - 2y = 6$ through (1,2) | $y = -\frac{2}{3}x + \frac{8}{3}$ |

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📌 Key Takeaways:

- Slope formula helps find steepness between two points.
- $y = mx + b$ is useful when you know slope and y-intercept.
- Point-slope form is best when you know a point and slope.
- Parallel lines → same slope.
- Perpendicular lines → slopes multiply to $-1$ (negative reciprocals).

This chart is excellent for reviewing linear equations! Let me know if you'd like a printable version or practice problems.