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

Let’s solve each problem one by one, using the slope-intercept form of a 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 is y = x + 4, which has slope m = 1.
- Parallel lines have the same slope, so our new line also has m = 1.
- Use point-slope form:
y - y₁ = m(x - x₁)
Plug in (x₁, y₁) = (2, 2), m = 1:
→ y - 2 = 1(x - 2)
→ y - 2 = x - 2
→ y = x

✔ Answer: y = x

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

- The line x = 0 is the y-axis, a vertical line.
- Vertical lines have undefined slope, and their equations are of the form x = constant.
- A line parallel to x = 0 must also be vertical → same x-value.
- Since it passes through (4, 3), the x-coordinate is 4 → x = 4

⚠️ But note: slope-intercept form (y = mx + b) does not apply to vertical lines, because they don’t have a defined slope or y-intercept.

However, since the problem asks for “slope-intercept form”, we must interpret this carefully. Technically, x = 4 cannot be written in slope-intercept form.

But perhaps the question expects us to recognize that and state it’s vertical — but let’s check if there’s an error.

Actually, maybe the problem meant “parallel to y = 0” (the x-axis)? That would make sense for slope-intercept form.

But as written: parallel to x = 0 → vertical line → x = 4

Since slope-intercept form doesn't exist for vertical lines, we can say:

✔ Answer: x = 4 (not in slope-intercept form, but this is the correct equation)

If forced to write in y = mx + b, it's impossible. So we report the correct line equation.

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

- Given line: slope m = 3
- Parallel → same slope: m = 3
- Point: (2, -4)
- Point-slope:
y - (-4) = 3(x - 2)
→ y + 4 = 3x - 6
→ y = 3x - 10

✔ Answer: y = 3x - 10

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4) Through (2, -1), parallel to y = -2/5 x + 3

- Slope of given line: m = -2/5
- Parallel → same slope: m = -2/5
- Point: (2, -1)
- Point-slope:
y - (-1) = (-2/5)(x - 2)
→ y + 1 = (-2/5)x + 4/5
→ y = (-2/5)x + 4/5 - 1
→ y = (-2/5)x - 1/5

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

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

- Slope of given line: m = 1/8
- Perpendicular lines have slopes that are negative reciprocals → new slope = -8
- Point: (1, -5)
- Point-slope:
y - (-5) = -8(x - 1)
→ y + 5 = -8x + 8
→ y = -8x + 3

✔ Answer: y = -8x + 3

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

- Slope of given line: m = 1
- Perpendicular slope: -1 (negative reciprocal)
- Point: (4, -1)
- Point-slope:
y - (-1) = -1(x - 4)
→ y + 1 = -x + 4
→ y = -x + 3

✔ Answer: y = -x + 3

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## ✔ Final Answers:

1) y = x
2) x = 4 *(Note: Not in slope-intercept form; vertical line)*
3) y = 3x - 10
4) y = -\frac{2}{5}x - \frac{1}{5}
5) y = -8x + 3
6) y = -x + 3

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📌 Important Note for #2: If your instructor expects *only* slope-intercept form, you may need to clarify whether "parallel to x=0" was a typo. Otherwise, x=4 is the correct geometric answer, even if it doesn’t fit y=mx+b.