Question type

Parallel and perpendicular lines on the SAT

Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: if one slope is $\frac{2}{5}$ , the other is $- \frac{5}{2}$ , and the two slopes multiply to $- 1$ . On the SAT, first find each slope, usually by rewriting standard form $a x + b y = c$ as $y = - \frac{a}{b} x + \frac{c}{b}$ .

These questions give you one line and ask about another that is parallel or perpendicular to it: its equation, a missing coordinate on it, or a constant in its equation.

It appeared on three of the five official practice tests we analysed. The rule itself is short; the points are lost in finding the slope of a line given in standard form.

On 3 of 5 official practice tests
Desmos helps

Why students miss it

Standard form hides the slope. The slope of $a x + b y = c$ is $- \frac{a}{b}$ , not $a$ and not $\frac{a}{b}$ . Reading the slope of $3 x - y = 9$ as $- 3$ instead of $3$ is the commonest slip.

For perpendicular lines, students flip the fraction and forget the sign, or change the sign and forget to flip. Both halves are needed: $\frac{2}{5}$ becomes $- \frac{5}{2}$ .

How to spot it

The words "parallel to" or "perpendicular to" next to a line's equation.
A line given in standard form, $a x + b y = c$ , beside one in slope-intercept form.
A missing coordinate, such as $(k, 11)$ , on a line with a stated relationship to another.

The method

Find the slope first

Rewrite every line as $y = m x + b$ , or use slope $= - \frac{a}{b}$ for $a x + b y = c$ .

Parallel: use the same slope.
Perpendicular: flip the fraction and change the sign.
Through two points: slope $= \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ , then set it equal.

With Desmos

Desmos draws both lines, so you can see at once whether a choice is parallel or perpendicular.

Type the given line exactly as written, standard form included: Desmos graphs 3x-y=9 directly.
For a missing constant, type the other line with the constant and add a slider.
Move the slider until the lines never meet (parallel) or meet at a right angle (perpendicular).

A right angle is hard to judge by eye. For perpendicular lines, check that the two slopes multiply to $- 1$ .

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#694Alg

In the $x y$ -plane, the line through $(2, 5)$ and $(k, 11)$ is parallel to the line defined by $3 x - y = 9$ . What is the value of $k$ ?

Pick an answer, then check it.

Show the answer and explanation

Rewrite

3 x - y = 9

y = 3 x - 9

, so the slope is

3

. Set the slope through the two points equal to it:

\frac{11 - 5}{k - 2} = 3

, so

k - 2 = 2

and

k = 4

2Question 2 of 2#695Alg

In the $x y$ -plane, the graph of $a x + 4 y = 9$ is perpendicular to the graph of $y = \frac{2}{5} x + 1$ , where $a$ is a constant. What is the value of $a$ ?

Pick an answer, then check it.

Show the answer and explanation

Perpendicular slope must be

- \frac{5}{2}

. Rewrite

a x + 4 y = 9

y = - \frac{a}{4} x + \frac{9}{4}

, so

- \frac{a}{4} = - \frac{5}{2}

, giving

a = 10

Questions students ask

What is the slope of a line perpendicular to y = 2/5 x + 1?

The negative reciprocal of 2/5, which is -5/2. Flip the fraction and change its sign; the two slopes multiply to -1.

How do you find the slope of a line in standard form?

For ax + by = c, the slope is -a/b. You can also solve for y: 3x - y = 9 becomes y = 3x - 9, so the slope is 3.

Do parallel lines have the same y-intercept?

No. Parallel lines have the same slope and different y-intercepts. With the same intercept as well, they are the same line.

Practise this type until it is automatic

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