SAT lines: find slope from any form, then spot parallel and perpendicular lines
Slope hides in every form of a linear equation. Learn to pull it out of standard form, point-slope form, or two points, and to recognize parallel and perpendicular lines instantly.
SPSAT Practice Editorial7 min read
The short version
What you'll take away
- Slope through two points is (y₂ − y₁) ÷ (x₂ − x₁); in Ax + By = C it is −A/B.
- Parallel lines have equal slopes and different intercepts.
- Perpendicular lines have slopes that are negative reciprocals: their product is −1.
In this article
Slope from every form
Slope is the rate of change: rise over run. The SAT presents lines in several forms, and each one hides the slope in a different place.
See the skill
Where the slope hides
Slope-intercept: y = mx + b
m
y = 3x − 2 has slope 3.
Standard: Ax + By = C
−A/B
4x − 2y = 6 has slope −4 ÷ (−2) = 2.
Point-slope: y − y₁ = m(x − x₁)
m
y − 5 = −2(x + 1) has slope −2.
Two points
(y₂ − y₁) ÷ (x₂ − x₁)
(1, 2) and (4, 11): 9 ÷ 3 = 3.
Parallel and perpendicular
Parallel lines rise at the same rate, so they have equal slopes. If their intercepts also match, they are the same line. Perpendicular lines meet at a right angle, and their slopes are negative reciprocals: flip the fraction and change the sign. A slope of 2 pairs with \(\displaystyle -\frac{1}{2}\), because \(\displaystyle 2 \times -\frac{1}{2} = -1\).
The study lab
Perpendicular lines
y = 2x + 1 and y = −½x + 1 meet at a right angle at (0, 1).
- y = 2x + 1 (slope 2)
- y = −½x + 1 (slope −½)
Practice: try these before checking the answers
Find each slope before comparing lines.
- 1. Find the slope of the line through (1, 2) and (4, 11).
- 2. Find the slope of \(\displaystyle 4x - 2y = 6\).
- 3. Write the line parallel to \(\displaystyle y = -3x + 5\) that passes through (0, 2).
- 4. What is the slope of a line perpendicular to \(\displaystyle y = 4x - 1\)?
- 5. Do \(\displaystyle 2x + 3y = 6\) and \(\displaystyle 4x + 6y = 5\) intersect?
Sources & further reading
One step from here
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