Math worksheet for writing equations of parallel and perpendicular lines, including steps and examples.

Let's solve the problem step by step, filling in the blanks and working through both examples. This worksheet is about writing equations of lines that are parallel or perpendicular to a given line and passing through a specific point.

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🔷 Key Concepts Recap

- Slope-Intercept Form: $ y = mx + b $
- $ m $ = slope
- $ b $ = y-intercept

- Parallel Lines: Same slope → $ m_{\text{parallel}} = m_{\text{original}} $

- Perpendicular Lines: Opposite reciprocal slopes
→ If original slope is $ m $, then perpendicular slope is $ -\frac{1}{m} $

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✔ Steps (Fill in the blanks)

1. Determine the slope of the parallel or perpendicular line
2. Plug in your $ x $ and $ y $ value in the given ordered pair
3. Substitute the $ x $, $ y $, and slope $ m $ into the slope-intercept form equation
4. Solve for $ b $
5. Substitute the slope, $ m $, and y-int, $ b $, into the slope-intercept form equation

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## 🟦 Example 1: Parallel Line

> Write the equation of a line that is parallel to $ y = 4x + 9 $ and passes through $ (0, 2) $

Step-by-step:

- Given line: $ y = 4x + 9 $ → slope $ m = 4 $
- Since it's parallel, the new line has the same slope: $ m = 4 $
- Point: $ (0, 2) $ → so $ x = 0 $, $ y = 2 $

Now plug into $ y = mx + b $:

$$
y = mx + b \\
2 = 4(0) + b \\
2 = 0 + b \\
b = 2
$$

So:
- $ m = 4 $
- $ b = 2 $

Final equation:
$$
y = 4x + 2
$$

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✔ Filled-in Example 1:

```
x = 0 y = 2 m = 4

y = mx + b
2 = 4(0) + b
2 = 0 + b
2 = b

m = 4 b = 2

y = mx + b
y = 4x + 2
```

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## 🟨 Example 2: Perpendicular Line

> Write the equation of a line that is perpendicular to $ y = -6x + 12 $ and passes through $ (12, -5) $

Step-by-step:

- Given line: $ y = -6x + 12 $ → slope $ m = -6 $
- For perpendicular, use opposite reciprocal:
- Reciprocal of $ -6 $ is $ -\frac{1}{6} $
- Opposite of that is $ \frac{1}{6} $
- So, $ m = \frac{1}{6} $

Point: $ (12, -5) $ → $ x = 12 $, $ y = -5 $

Now plug into $ y = mx + b $:

$$
y = mx + b \\
-5 = \frac{1}{6}(12) + b \\
-5 = 2 + b \\
b = -5 - 2 = -7
$$

So:
- $ m = \frac{1}{6} $
- $ b = -7 $

Final equation:
$$
y = \frac{1}{6}x - 7
$$

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✔ Filled-in Example 2:

```
x = 12 y = -5 m = 1/6

y = mx + b
-5 = (1/6)(12) + b
-5 = 2 + b
-7 = b

m = 1/6 b = -7

y = mx + b
y = (1/6)x - 7
```

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✔ Final Answers:

#### Example 1 (Parallel):
- Equation: $ \boxed{y = 4x + 2} $

#### Example 2 (Perpendicular):
- Equation: $ \boxed{y = \frac{1}{6}x - 7} $

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📌 Summary

| Type | Slope Rule | Example Slope |
|-------------|------------------------|---------------|
| Parallel | Same slope | $ m = 4 $ |
| Perpendicular | Opposite reciprocal | $ m = \frac{1}{6} $ |

Always:
1. Use the correct slope based on relationship.
2. Plug in the point $ (x, y) $ to find $ b $.
3. Write final equation in $ y = mx + b $ form.

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