Work with parallel and perpendicular lines

Most of the time, the question says parallel or perpendicular outright. Watch for these other ways of saying it too:

• A line that has to have the same slope as another line is the parallel case.
• Two slopes that multiply to $-1$ belong to perpendicular lines.
• A missing number, like the $k$ in $kx-10y=7$ or in the point $(k,-5)$, is sometimes there to make two lines parallel or perpendicular.

Whatever the wording, the plan has three steps:

1. Find the slope of the line you’re given.
2. Change that slope to fit the relationship. Keep it for parallel, and take the negative reciprocal for perpendicular.
3. Use whatever is left, like a point, a coefficient or a coordinate, to finish.

Here are the two rules, with $m_1$ and $m_2$ for the two slopes. They work whenever neither line is vertical, because a vertical line’s slope is undefined. Horizontal and vertical lines get their own rules a little further on.

Parallel lines

Parallel lines have the same slope:

For example, every line parallel to

has slope $-\frac45$. It can cross the $y$-axis somewhere else, but it points the same way.

One catch: two equations with the same slope might describe two parallel lines, or they might be the same line. If a question wants a different parallel line, its intercept has to be different.

Perpendicular lines

Perpendicular slopes multiply to $-1$:

Why does the fraction flip? Look back at the quarter turn in the SAT example: right $3$, down $2$ became right $2$, up $3$. The $3$ and the $2$ traded places. That’s the flip. Down became up. That’s the sign change.

Every slope works the same way. A slope of $\frac ab$ means right $b$, up $a$. A quarter turn makes it right $a$, down $b$, which is a slope of $-\frac ba$. So for any slope that isn’t $0$, if

then

That’s the negative reciprocal, and it comes down to one phrase worth remembering: flip the fraction, flip the sign.

• The negative reciprocal of $\frac35$ is $-\frac53$.
• The negative reciprocal of $-\frac35$ is $\frac53$.
• For a whole number, write it as a fraction first. The negative reciprocal of $4=\frac41$ is $-\frac14$.

To check your answer, multiply. You should get $-1$:

Lots of SAT lines come in standard form,

As long as $B\ne0$, which means the line isn’t vertical, solve for $y$:

So the slope is

The constant $C$ slides the line around but never changes its slope. That’s why

and

are parallel. The coefficients $4$ and $6$ are double $2$ and $3$, so both lines have slope $-\frac23$. They aren’t the same line, though: doubling the first equation gives $4x+6y=12$, not $15$.

For perpendicular lines, find both slopes and set their product equal to $-1$. Say

is perpendicular to

The slopes are $-\frac52$ and $\frac{k}{10}$, so

Working it out like this is usually faster than graphing, and it’s exact. On an ordinary graph, two lines that are nearly parallel can look exactly parallel.

A vertical line has no slope to put into $m_1m_2=-1$, so here you work from the equations instead.

The grid on graph paper shows all three facts at once:

• Horizontal lines are parallel to other horizontal lines.
• Vertical lines are parallel to other vertical lines.
• Every horizontal line is perpendicular to every vertical line.

Say a line passes through $(-3,2)$ and is perpendicular to

The line $y=6$ is horizontal, so the new line has to be vertical. The vertical line through $(-3,2)$ is

Every point on a vertical line has the same $x$-coordinate, so the point’s $x$-coordinate, $-3$, is all you need. Its $y$-coordinate, $2$, doesn’t appear in the equation at all.