Writing Equations of Parallel and Perpendicular Lines Notes ... - Free Printable

Let's solve the problem step by step, filling in the blanks and working through both examples. The goal is to write equations of lines that are either parallel or perpendicular to a given line and pass 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 $)
- Perpendicular Lines: Slopes are opposite reciprocals
- If one slope is $ m $, the 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 $ 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 1: Find the slope

- Since it's parallel, the slope $ m $ is the same as the original line.
- Original slope = $ 4 $
- So, $ m = 4 $

Step 2: Use the point $ (0, 2) $

- $ x = 0 $, $ y = 2 $

Step 3: Plug into $ y = mx + b $

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

Step 4: Write the equation

- $ m = 4 $, $ b = 2 $
- Equation: $ y = 4x + 2 $

✔ Final Answer: $ \boxed{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 1: Find the slope

- Original slope = $ -6 $
- Perpendicular slope = opposite reciprocal → $ \frac{1}{6} $
- So, $ m = \frac{1}{6} $

Step 2: Use the point $ (12, -5) $

- $ x = 12 $, $ y = -5 $

Step 3: Plug into $ y = mx + b $

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

Step 4: Write the equation

- $ m = \frac{1}{6} $, $ b = -7 $
- Equation: $ y = \frac{1}{6}x - 7 $

✔ Final Answer: $ \boxed{y = \frac{1}{6}x - 7} $

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✔ Filled-in Worksheet:

#### Example 1 (Parallel):
```
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
```

#### Example 2 (Perpendicular):
```
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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✔ Summary:

- Parallel: Same slope → use $ m = 4 $, find $ b $
- Perpendicular: Opposite reciprocal slope → $ m = \frac{1}{6} $, then find $ b $

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