How to Write Equation of Parallel and Perpendicular Lines?
Parallel and Perpendicular Lines
Two slope facts unlock this topic: parallel lines have the same slope, and perpendicular lines have slopes that are negative reciprocals (their product is \(-1\)). With those, you can write a line parallel or perpendicular to any given line. We’ll do both, with a solver, practice, and a worksheet maker a tap away.
Two lines’ slopes tell you exactly how they relate. Parallel lines never meet, and that’s because they share the same slope. Perpendicular lines cross at a right angle, and their slopes are negative reciprocals — flip the fraction and change the sign. Master those two facts and you can write a line parallel or perpendicular to any line, through any point.
In short: parallel lines have equal slopes (\(m_1 = m_2\)); perpendicular lines have slopes whose product is \(-1\) (\(m_2 = -\tfrac{1}{m_1}\)). A line with slope 2 is parallel to other slope-2 lines and perpendicular to slope \(-\tfrac12\) lines.
Same Slope vs. Negative Reciprocal
Parallel lines climb at the same rate, so they keep the same distance apart forever — equal slopes. Perpendicular lines meet at \(90°\); turning a direction a right angle flips rise and run and reverses the sign, which is exactly the negative reciprocal. Their slopes multiply to \(-1\).
How to find the new slope:
- Parallel: use the same slope as the given line.
- Perpendicular: flip the slope and change its sign (negative reciprocal).
- Then use point-slope with the given point to write the equation.
Perpendicular slopes
\(y = 2x + 1\) (slope 2) and \(y = -\tfrac12 x + 2\) (slope \(-\tfrac12\)) meet at a right angle, because \(2 \times -\tfrac12 = -1\). A parallel line to the first would also have slope 2.
⚡ Build a related lineWorked Examples
Same slope stays parallel; the negative reciprocal turns a right angle — both pairs are drawn below.
Example A — Parallel line
Write the line parallel to \(y = 2x + 1\) through \((0, -3)\).
- Parallel means the same slope: \(m = 2\).
- The point \((0,-3)\) gives the intercept \(b = -3\).
- Write it: \(y = 2x – 3\).
Answer: \(y = 2x – 3\)
Example B — Perpendicular line
Write the line perpendicular to \(y = 2x + 1\) through \((0, 4)\).
- Negative reciprocal of 2: flip to \(\tfrac12\), change sign to \(-\tfrac12\).
- The point \((0,4)\) gives \(b = 4\).
- Write it: \(y = -\tfrac12 x + 4\).
Answer: \(y = -\tfrac12 x + 4\)
Example C — Perpendicular to a negative slope
Find the slope perpendicular to \(y = -3x + 5\).
- Write \(-3\) as \(-\tfrac31\).
- Flip and change sign: \(\tfrac13\).
- Check: \(-3 \times \tfrac13 = -1\) ✓.
Answer: \(\tfrac13\)
Example D — Use point-slope
Perpendicular to \(y = \tfrac14 x\) through \((2, 1)\).
- Negative reciprocal of \(\tfrac14\) is \(-4\).
- Point-slope: \(y – 1 = -4(x – 2)\).
- Simplify: \(y = -4x + 9\).
Answer: \(y = -4x + 9\)
Where You’ll Use It
Parallel and perpendicular relationships are everywhere in geometry and design: the opposite sides of a rectangle are parallel; a road meeting another at a right angle is perpendicular. In coordinate geometry, you use these slope rules to prove shapes, find shortest distances, and build perpendicular bisectors.
Slip-Ups That Cost Easy Points
- Forgetting the sign for perpendicular. It’s the negative reciprocal — flip and change the sign. The perpendicular of 2 is \(-\tfrac12\), not \(\tfrac12\).
- Only flipping, or only negating. You must do both. \(-\tfrac34 \to +\tfrac43\).
- Changing the slope for a parallel line. Parallel keeps the same slope; only the intercept differs.
- Special cases. A horizontal line (slope 0) is perpendicular to a vertical line (undefined slope) — the reciprocal rule doesn’t apply numerically there.
Your Turn
Find each requested slope or equation, then reveal.
- Slope parallel to \(y = -5x + 2\)?
- Slope perpendicular to \(y = 4x\)?
- Line parallel to \(y = 3x + 1\) through \((0, -2)\)?
- Slope perpendicular to \(y = \tfrac23 x – 1\)?
Show answers
- \(\color{blue}{-5}\)
- \(\color{blue}{-\tfrac14}\)
- \(\color{blue}{y = 3x – 2}\)
- \(\color{blue}{-\tfrac32}\)
Make Your Own Worksheet
Generate fresh parallel/perpendicular problems with a full answer key — print or save as a PDF.
Frequently Asked Questions
How are the slopes of parallel lines related?
They’re equal. Parallel lines rise at the same rate, so \(m_1 = m_2\); only their y-intercepts differ.
How are the slopes of perpendicular lines related?
They’re negative reciprocals — flip the fraction and change the sign, so their product is \(-1\). The perpendicular of \(2\) is \(-\tfrac12\).
How do I write the equation once I have the slope?
Use the new slope with the given point in point-slope form, \(y – y_1 = m(x – x_1)\), then simplify to \(y = mx + b\).
What about horizontal and vertical lines?
They’re perpendicular to each other: a horizontal line has slope 0 and a vertical line has undefined slope. The negative-reciprocal formula doesn’t apply numerically in that special case.
Related Topics
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