Linear equations in two variables: graph and interpret lines, find slope and intercepts, convert between slope-intercept and standard form, and find the equation of a line from given information.

What this skill is asking

A linear equation in two variables describes a line in the $xy$-plane. The Digital SAT (Algebra domain) tests fluency moving among the three standard forms, reading slope and intercepts, using parallel and perpendicular slope relationships, and finding a line's equation from points or other clues. This is the geometric heart of the Algebra domain.

The three forms

Each form is best for a different task.

A worked equation-building question

Finding a line's equation is a core SAT task.

Slope, intercepts and what they mean

The slope is the rate at which $y$ changes per unit increase in $x$; a steeper line has a larger absolute slope. The $y$-intercept is where the line crosses the $y$-axis ($x = 0$), and the $x$-intercept is where it crosses the $x$-axis ($y = 0$). On the SAT these often carry a context, for example a cost line whose $y$-intercept is a fixed fee and whose $x$-intercept is a break-even point. Standard form $Ax + By = C$ is convenient because the intercepts fall out immediately: set $y = 0$ to get $x = \frac{C}{A}$, and set $x = 0$ to get $y = \frac{C}{B}$.

Parallel and perpendicular lines

These slope relationships appear constantly. Parallel lines never meet, so they share the same slope but have different intercepts. Perpendicular lines meet at a right angle, and their slopes are negative reciprocals: flip the fraction and change the sign, so $3$ pairs with $-\frac{1}{3}$ and $-\frac{2}{5}$ pairs with $\frac{5}{2}$. A quick check is that the product of perpendicular slopes is $-1$. The Desmos calculator can confirm any of this visually: graph the lines and look for the same direction (parallel) or a right-angle crossing (perpendicular).

Two edge cases are worth knowing because they break the negative-reciprocal rule. A horizontal line has slope $0$ and equation $y = k$; a vertical line has an undefined slope and equation $x = h$. A horizontal and a vertical line are perpendicular to each other, but you cannot get there by taking a negative reciprocal of $0$ (which is undefined). So when a question involves a horizontal or vertical line, reason from the picture (horizontal is flat, vertical is upright) rather than from the slope formula. Recognising $y = 3$ as a horizontal line and $x = -2$ as a vertical line, and that any horizontal line is perpendicular to any vertical line, handles these without algebra.