Equations of Lines: Slope, Parallel, and Perpendicular — SAT Math

Introduction

Linear equations in two variables appear throughout the SAT's Algebra domain, which makes up about 35% of the math section. This note focuses on one precise skill: writing the equation of a line when you're given two points, one point and a slope, or a parallel/perpendicular relationship. You'll also move fluidly between slope-intercept, point-slope, and standard form, and read key features (intercepts, slope) directly from each.

For real-world applications of these ideas — rate of change, unit interpretation — see the sibling note Linear Equations in Two Variables in Context.

Core Concept

Every non-vertical line in the $xy$ -plane is completely determined by its slope and one point on it. Everything else follows from the slope formula.

The Slope Formula

Given two points $(x_1, y_1)$ and $(x_2, y_2)$ :

m = \frac{y_2 - y_1}{x_2 - x_1}

Slope measures rise over run. It is constant across any two points on a line — a fact that makes tables easy to check: equal spacing in $x$ must produce equal spacing in $y$ .

The Three Forms

Form	Equation	Best used when…
Slope-intercept	$y = mx + b$	You know slope $m$ and $y$ -intercept $b$
Point-slope	$y - y_1 = m(x - x_1)$	You know slope $m$ and any point $(x_1, y_1)$
Standard	$Ax + By = C$	Answer choices or the problem are in this form

All three forms represent the same line; converting between them is just algebra.

Reading Intercepts

$y$ -intercept: set $x = 0$ and solve for $y$ .
$x$ -intercept: set $y = 0$ and solve for $x$ .

From $Ax + By = C$ : $y$ -intercept $= C/B$ , $x$ -intercept $= C/A$ , slope $= -A/B$ .

Parallel and Perpendicular

Parallel lines have equal slopes: $m_1 = m_2$ .
Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$ , or equivalently $m_2 = -\dfrac{1}{m_1}$ .

A horizontal line ( $m = 0$ ) is perpendicular to a vertical line (undefined slope), but those special cases rarely appear in the equation-writing problems tested here.

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