Math Core

Railroad tracks never meet, and the corner of a sheet of paper is a perfect right angle. Both of these relationships, parallel and perpendicular, show up in the slopes of lines. Once you know the rule, you can tell how two lines meet just by looking at their equations.

Parallel lines

Two lines in a plane are parallel if they never intersect. For that to happen they must rise at exactly the same rate. If one line were even slightly steeper, it would eventually catch up to the other and cross it.

Three parallel lines with slope 1/2 and y-intercepts 3, −1 and −4.Open in grapher →

Slopes of parallel lines

Two different non-vertical lines are parallel exactly when they have the same slope.

All vertical lines are parallel to each other, too (their slopes are all undefined).

The word "different" matters. The equations $y = 2x + 1$ and $2y = 4x + 2$ have the same slope, but they're the same line written two ways (the second simplifies to the first). Parallel lines have the same slope and different $y$ -intercepts.

Perpendicular lines

Two lines are perpendicular if they meet at a right angle. Their slopes are related in a less obvious way.

Picture the line with slope $\dfrac{2}{3}$ : its slope triangle goes right $3$ and up $2$ . Now rotate that triangle a quarter turn counterclockwise. The run of $3$ becomes a rise of $3$ , and the rise of $2$ becomes a run of $2$ to the left. The rotated line has slope $\dfrac{3}{-2} = -\dfrac{3}{2}$ .

The lines y = (2/3)x and y = −(3/2)x meet at a right angle at the origin.Open in grapher →

Rotating by a right angle flipped the fraction and changed the sign. That is the rule.

Slopes of perpendicular lines

Two non-vertical lines are perpendicular exactly when their slopes are opposite reciprocals:

m_2 = -\frac{1}{m_1}, \qquad\text{or equivalently}\qquad m_1 \cdot m_2 = -1.

A horizontal line (slope $0$ ) and a vertical line (undefined slope) are also perpendicular.

To find a perpendicular slope, flip the fraction and change the sign.

slope	perpendicular slope	check: product
$\dfrac{2}{3}$	$-\dfrac{3}{2}$	$-1$
$4$	$-\dfrac{1}{4}$	$-1$
$-\dfrac{1}{5}$	$5$	$-1$
$-1$	$1$	$-1$

Common mistake

Do both steps. Flipping $\dfrac{2}{3}$ to $\dfrac{3}{2}$ without changing the sign gives a line that is not perpendicular (the product is $1$ , not $-1$ ). Changing the sign without flipping, $-\dfrac{2}{3}$ , is also wrong.

Classifying pairs of lines

To compare two lines, put both in slope-intercept form and compare slopes.

Worked example: Parallel, perpendicular or neither?

$y = 3x - 4$ and $y = 3x + 2$
$y = -\dfrac{1}{2}x + 1$ and $y = 2x - 5$
$3x + y = 5$ and $6x + 2y = 7$
$y = 4x + 1$ and $y = -4x + 1$

Solutions.

Both slopes are $3$ and the intercepts differ: parallel.
$-\dfrac{1}{2} \cdot 2 = -1$ : perpendicular.
Solve each for $y$ : $y = -3x + 5$ and $y = -3x + \dfrac{7}{2}$ . Same slope, different intercepts: parallel.
The slopes are $4$ and $-4$ . Their product is $-16$ , not $-1$ , and they aren't equal: neither. (Opposite slopes are not enough for perpendicular lines; you also need reciprocals.)

Writing equations of parallel and perpendicular lines

A typical question: "Write the equation of the line through a given point that is parallel (or perpendicular) to a given line." The plan is always the same.

Find the slope of the given line.
Use the same slope (parallel) or the opposite reciprocal (perpendicular).
Use point-slope form with the given point, then simplify.

Worked example: A parallel line

Write the equation of the line through $(2, 9)$ parallel to $y = 3x - 1$ .

The given slope is $3$ , so the new slope is also $3$ .

y - 9 = 3(x - 2) \;\Longrightarrow\; y - 9 = 3x - 6 \;\Longrightarrow\; y = 3x + 3.

Check: $3(2) + 3 = 9$ , and the intercept $3$ differs from $-1$ , so the lines really are parallel and distinct.

Worked example: A perpendicular line

Write the equation of the line through $(4, 1)$ perpendicular to $2x + 5y = 10$ .

Given slope. Solve for $y$ : $5y = -2x + 10$ , so $y = -\dfrac{2}{5}x + 2$ . The slope is $-\dfrac{2}{5}$ .

New slope. Flip and change the sign: $\dfrac{5}{2}$ .

Equation.

\begin{aligned} y - 1 &= \frac{5}{2}(x - 4) \\ y - 1 &= \frac{5}{2}x - 10 \\ y &= \frac{5}{2}x - 9 \end{aligned}

Check: $-\dfrac{2}{5} \cdot \dfrac{5}{2} = -1$ , and $\dfrac{5}{2}(4) - 9 = 10 - 9 = 1$ .

Tip

Before you finish, multiply the two slopes. Parallel lines give equal slopes; perpendicular lines give a product of exactly $-1$ . It takes five seconds and catches most sign errors.

Practice

Practice 1

What is the slope of any line parallel to $y = -7x + 2$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

What is the slope of any line perpendicular to $y = \dfrac{3}{4}x - 1$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

How are the lines $y = 2x + 1$ and $y = -\dfrac{1}{2}x + 4$ related?

Practice 4

How are the lines $x - 3y = 6$ and $y = 3x + 2$ related?

Practice 5

Write the equation of the line through $(-1, 2)$ that is parallel to $y = 4x - 3$ . Use slope-intercept form.

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 6

Write the equation of the line through $(6, 1)$ that is perpendicular to $y = -3x + 8$ . Use slope-intercept form.

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 7

Write the equation of the line through $(8, -3)$ that is perpendicular to $4x - 5y = 20$ . Use slope-intercept form.

Enter an expression, e.g. 3x^2 - 2x + 1

Practice 8

For what value of $k$ is the line $y = kx + 2$ perpendicular to the line $y = \dfrac{2}{3}x - 5$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.