Equations of Perpendicular Lines

Q: How do I find the perpendicular slope of a whole number like $4$?

Write it as a fraction over $1$ first: $4 = \dfrac{4}{1}$. Flip it to $\dfrac{1}{4}$, then change the sign to get $-\dfrac{1}{4}$. Verify with the product test: $4 \cdot \left(-\dfrac{1}{4}\right) = -1$.

Two lines are parallel if they never cross, and perpendicular if they cross at a right angle. On a coordinate plane, both relationships come down to one number: slope. Parallel lines have the same slope. Perpendicular lines have slopes that are negative reciprocals of each other — flip the fraction, flip the sign.

That's the entire topic. Every problem in this lesson — checking whether two lines are parallel, writing a line perpendicular to a given one through a given point — starts by finding a slope and applying one of those two rules.

Parallel lines: same slope

Slope is the direction a line points. Two lines with the same slope climb (or fall) at exactly the same rate, so the vertical gap between them never changes — they can't ever meet. The lines $y = 3x + 2$ and $y = 3x - 5$ are parallel: same slope $3$ , different $y$ -intercepts.

The $y$ -intercepts must be different. If two equations have the same slope and the same $y$ -intercept, they are the same line, not parallel lines. So the parallel check is: slopes equal, intercepts not equal.

Perpendicular lines: negative reciprocal slopes

Two lines are perpendicular when their slopes multiply to $-1$ . Equivalently, each slope is the negative reciprocal of the other: flip the fraction and change the sign. A line with slope $\dfrac{3}{4}$ is perpendicular to a line with slope $-\dfrac{4}{3}$ , because $\dfrac{3}{4} \cdot \left(-\dfrac{4}{3}\right) = -1$ .

For a whole-number slope, write it over $1$ first. Perpendicular to slope $2$ ? Write $2$ as $\dfrac{2}{1}$ , flip to $\dfrac{1}{2}$ , change the sign: $-\dfrac{1}{2}$ . Check: $2 \cdot \left(-\dfrac{1}{2}\right) = -1$ .

The graph below shows a perpendicular pair: the black line $y = 2x - 1$ and the blue line $y = -\dfrac{1}{2}x + 2$ . The slopes are $2$ and $-\dfrac{1}{2}$ , their product is $-1$ , and the lines cross at a right angle.

Horizontal and vertical lines

Horizontal lines like $y = 3$ have slope $0$ . Vertical lines like $x = 2$ have undefined slope. The negative-reciprocal rule can't be applied to them — you can't flip $\dfrac{0}{1}$ into anything usable — but the geometry still works: every horizontal line is perpendicular to every vertical line.

So treat them as a memorized special case: parallel to a horizontal line means horizontal; perpendicular to a horizontal line means vertical, and vice versa.

Worked examples

Example 1: slope of a parallel line

What is the slope of a line parallel to $y = -3x + 7$ ?

Read the slope of the given line

m = -3

Parallel lines have the same slope

m_{\text{parallel}} = -3

Answer: $-3$

Example 2: slope of a perpendicular line

What is the slope of a line perpendicular to a line with slope $\dfrac{2}{5}$ ?

Flip the fraction

\dfrac{2}{5} \rightarrow \dfrac{5}{2}

Change the sign

m_{\perp} = -\dfrac{5}{2}

Check the product

\dfrac{2}{5} \cdot \left(-\dfrac{5}{2}\right) = -1

Answer: $-\dfrac{5}{2}$

Example 3: write a parallel line through a point

Write the equation of the line parallel to $y = 2x + 3$ through $(1, 4)$ .

Parallel slope matches the given line

m = 2

Use point-slope form with

(1, 4)

y - 4 = 2(x - 1)

Distribute

y - 4 = 2x - 2

Add

4

to both sides

y = 2x + 2

Check the point:

2(1) + 2 = 4

, so

(1, 4)

is on the line

Answer: $y = 2x + 2$

Example 4: write a perpendicular line through a point

Write the equation of the line perpendicular to $y = 3x - 1$ through $(6, 2)$ .

Slope of the given line

m_1 = 3

Take the negative reciprocal

m_2 = -\dfrac{1}{3}

Use point-slope form with

(6, 2)

y - 2 = -\dfrac{1}{3}(x - 6)

Distribute

y - 2 = -\dfrac{1}{3}x + 2

Add

2

to both sides

y = -\dfrac{1}{3}x + 4

Check: slopes multiply to

3 \cdot \left(-\dfrac{1}{3}\right) = -1

, and

-\dfrac{1}{3}(6) + 4 = 2

Answer: $y = -\dfrac{1}{3}x + 4$

Try one yourself

Common questions

If two lines have the same slope, are they always parallel?

Almost — there's one exception. If they also share the same $y$ -intercept, they are the same line, not two parallel lines. Same slope plus different intercepts means parallel.

How do I find the perpendicular slope of a whole number like

4

Write it as a fraction over $1$ first: $4 = \dfrac{4}{1}$ . Flip it to $\dfrac{1}{4}$ , then change the sign to get $-\dfrac{1}{4}$ . Verify with the product test: $4 \cdot \left(-\dfrac{1}{4}\right) = -1$ .

Do horizontal and vertical lines follow the multiply-to-

-1

rule?

No — a horizontal line has slope $0$ and a vertical line has undefined slope, so the product $-1$ test can't be computed. They are still perpendicular to each other; it's a special case you handle by inspection rather than by formula.

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