Digital SAT Math · Algebra

Graphs of linear equations

The picture is already the solution. When the test draws a line or a shaded half-plane, the fastest path is not algebra first — it is two features read off the grid: the sign of the slope and the y-intercept. Those two kills usually leave one choice standing before you ever count rise over run.

On the test

Recognition cues: in the xy-plane; a graph with two marked lattice points; shaded region; solid or dashed boundary; answer choices that are four versions of $y = mx + b$ or four inequalities that differ only in a symbol or a sign.

Where this skill ends and its neighbours begin

Algebra-first work on one line — points to equation, standard-form slope $-\frac{a}{b}$, parallel lines — lives on linear equations in two variables. Function language $f(x)$, tables, input-vs-output lives on linear functions. One-variable inequalities and word limits live on linear inequalities. This page is picture-first: the stem hands you a graph (or asks you to treat intercepts as a graph), and the job is to read it. Two boundaries at once is systems of linear inequalities — not here.

Pattern recognition

Sort by what the picture gives you:

1. Match the equation — kill by sign of slope, then by y-intercept, then confirm rise/run on lattice points.
2. Read one number — slope only, or one intercept only; leave the other feature alone.
3. Match the inequality — boundary equation first, solid/dashed next, test point last.
4. Point in / out of a half-plane — substitute; strict symbols exclude the boundary.
5. Intercepts from an equation — set the other variable to 0; do not rearrange the whole line unless asked.

Method

1. Read intercepts off the picture. Where the line meets the vertical axis is $b$. Where it meets the horizontal axis is the x-intercept. Only trust crossings that land on lattice points (or that the stem labels).
2. Slope by rise/run on lattice points only. Pick two points the line passes through exactly. $m = \dfrac{y_2 - y_1}{x_2 - x_1}$. Never estimate between grid lines.
3. Match sign-of-slope first to kill choices. Rising → positive; falling → negative. Then match $b$. What remains is usually one equation.
4. For inequalities: boundary style, then a test point. Solid means inclusive ($\le$ or $\ge$); dashed means strict ($<$ or $>$). Pick any labeled point in the shaded region and substitute — the side that makes the inequality true is the answer.

Slope and intercept, from a picture

Practice

Answer before you open the explanation. Six items depend on a figure — the numbers you need are on the grid, not in the sentence. Two items are student-produced response. Every wrong choice below is a specific, named error: inverted rise/run, intercept swap, sign of slope, solid-vs-dashed, or shading without a test.

Common mistakes

1. Swapping rise and run — computing $\frac{\Delta x}{\Delta y}$. The reciprocal often sits in the choices with the correct sign, which is why it survives a glance at “direction.”
2. Shading without a test point — assuming “above always means greater” when the inequality is written in a rearranged form, or when the figure is the only evidence.
3. Strict vs inclusive from the wrong line style — solid read as dashed (or the reverse), which flips $\le$ into $<$ and changes whether boundary points count.
4. Reporting the x-intercept as the y-intercept — both numbers appear in the stem or on the axes; the question names one of them.
5. Using a marked point’s y-coordinate as $b$ when that point is not on the vertical axis. The intercept is the height at $x = 0$.
6. Ignoring the sign of the slope — matching steepness only, so a falling line is paired with $+\frac{3}{2}$ instead of $-\frac{3}{2}$.
7. Estimating non-lattice crossings — reading “about 2.3” off the grid turns an exact slope into a number that will not appear among the choices.
8. Mismatched subtraction order — $\dfrac{y_2 - y_1}{x_1 - x_2}$. Both differences must run the same direction; reverse one and you only flip the sign.
9. Treating a boundary point as a solution of a strict inequality — $(0, 3)$ does not satisfy $y < -2x + 3$.

FAQ

Do I always need two points to match an equation? You need two independent facts. Sign of slope + y-intercept often leaves one choice; rise/run on a lattice pair confirms it. If the intercept is not lattice-clear, harvest two lattice points and compute both $m$ and $b$.

Solid vs dashed — which symbols go with which? Solid ↔ inclusive ($\le, \ge$). Dashed ↔ strict ($<, >$). The side (above/below) is a separate decision settled by a test point.

Can I use Desmos instead of counting squares? Yes for confirmation. For a single labeled lattice pair, counting is usually faster than typing. Use Desmos when you want to overlay a candidate equation on the picture you already read.

What if the line is vertical or horizontal? Horizontal means $m = 0$, equation $y = k$. Vertical means undefined slope, equation $x = h$ — rare on the Digital SAT but worth recognizing so you do not force a $y = mx + b$ form that cannot fit.

Is a grid-in slope allowed to be negative or fractional? Both. Enter -3/2 exactly rather than a rounded decimal.

Editorial team · last reviewed 2026-08-12