1.4 Ws - Equ of Perpendicular and Parallel Lines | PDF - Free Printable

Let's solve each problem step by step, writing the slope-intercept form of the line:
$$
y = mx + b
$$
where $ m $ is the slope and $ b $ is the y-intercept.

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1) Through: (2, 2), parallel to $ y = x + 4 $

- The given line $ y = x + 4 $ has slope $ m = 1 $.
- Parallel lines have the same slope, so our line also has slope $ m = 1 $.
- Use point-slope form: $ y - y_1 = m(x - x_1) $
$$
y - 2 = 1(x - 2)
$$
$$
y - 2 = x - 2
$$
$$
y = x
$$

✔ Answer: $ \boxed{y = x} $

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2) Through: (4, 3), parallel to $ x = 0 $

- $ x = 0 $ is a vertical line (the y-axis).
- Lines parallel to vertical lines are also vertical.
- A vertical line through $ (4, 3) $ has equation $ x = 4 $.
- But vertical lines do not have slope-intercept form because they are not functions (no defined slope).

⚠️ However, the question asks for slope-intercept form, which only applies to non-vertical lines. Since this line is vertical, it cannot be written in slope-intercept form.

But let’s check: if we must write an equation, we say:
$$
x = 4
$$
This is not in slope-intercept form.

✘ So, no slope-intercept form exists for this line.

But perhaps the intent is to recognize that vertical lines are not expressible in $ y = mx + b $ form.

✔ Answer: $ \boxed{x = 4} $ — not in slope-intercept form (so technically, no slope-intercept form exists).

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3) Through: (2, -4), parallel to $ y = 3x + 2 $

- Given line has slope $ m = 3 $
- Parallel ⇒ same slope: $ m = 3 $
- Point-slope form:
$$
y - (-4) = 3(x - 2)
$$
$$
y + 4 = 3x - 6
$$
$$
y = 3x - 10
$$

✔ Answer: $ \boxed{y = 3x - 10} $

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4) Through: (2, -1), parallel to $ y = -\frac{2}{5}x + 3 $

- Slope of given line: $ m = -\frac{2}{5} $
- Parallel ⇒ same slope: $ m = -\frac{2}{5} $
- Use point-slope:
$$
y - (-1) = -\frac{2}{5}(x - 2)
$$
$$
y + 1 = -\frac{2}{5}x + \frac{4}{5}
$$
$$
y = -\frac{2}{5}x + \frac{4}{5} - 1
$$
$$
y = -\frac{2}{5}x - \frac{1}{5}
$$

✔ Answer: $ \boxed{y = -\frac{2}{5}x - \frac{1}{5}} $

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5) Through: (1, -5), perpendicular to $ y = \frac{1}{8}x + 2 $

- Given line has slope $ m = \frac{1}{8} $
- Perpendicular lines have negative reciprocal slopes.
$$
m_{\perp} = -\frac{1}{(1/8)} = -8
$$
- Now use point-slope with $ m = -8 $, point $ (1, -5) $:
$$
y - (-5) = -8(x - 1)
$$
$$
y + 5 = -8x + 8
$$
$$
y = -8x + 3
$$

✔ Answer: $ \boxed{y = -8x + 3} $

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6) Through: (4, -1), perpendicular to $ y = x + 2 $

- Given line has slope $ m = 1 $
- Perpendicular slope is negative reciprocal: $ m = -1 $
- Use point-slope:
$$
y - (-1) = -1(x - 4)
$$
$$
y + 1 = -x + 4
$$
$$
y = -x + 3
$$

✔ Answer: $ \boxed{y = -x + 3} $

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✔ Final Answers:

1) $ y = x $
2) $ x = 4 $ — not in slope-intercept form
3) $ y = 3x - 10 $
4) $ y = -\frac{2}{5}x - \frac{1}{5} $
5) $ y = -8x + 3 $
6) $ y = -x + 3 $

> Note: For #2, since the line is vertical, it cannot be written in slope-intercept form. If required, you may state that "no slope-intercept form exists" or write $ x = 4 $ as the equation.