Math · College algebra · Worked example

Write parallel and perpendicular lines

Find the lines through (3, −1) that are parallel and perpendicular to 2x − 3y = 6.

Find the given slope

Solve 2x − 3y = 6 for y: −3y = −2x + 6, so y = (2x − 6)/3, which simplifies to slope-intercept form with slope 2/3.

\begin{array}{l} \frac{2 x - 6}{3} \\ \frac{2}{3} x - 2 \end{array}

A parallel line has the same slope, 2/3. Start from (3, −1).

\begin{array}{l} y = - 1 + \frac{2}{3} (x - 3) \\ y = \frac{2}{3} x - 3 \end{array}

Take the negative reciprocal of 2/3. The two slopes multiply to −1.

\frac{2}{3} \cdot (- \frac{3}{2}) = - 1

Start from (3, −1) with slope −3/2.

\begin{array}{l} y = - 1 - \frac{3}{2} (x - 3) \\ y = - \frac{3}{2} x + \frac{7}{2} \end{array}

Parallel: y = (2/3)x − 3. Perpendicular: y = −(3/2)x + 7/2.

Find the line through (−4, 2) that is perpendicular to y = 4x − 1.

Show the answer and explanation

The perpendicular slope is −1/4. Then y = 2 − (1/4)(x + 4) = −(1/4)x + 1.

\begin{array}{l} y = 2 - \frac{1}{4} (x + 4) \\ y = - \frac{1}{4} x + 1 \end{array}

Graph draws the given line, the parallel line and the perpendicular line through (3, −1).

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