Linear Equations & Systems

Linear equations are the largest topic in the Digital SAT Math section ("Algebra"). Master them and you secure a big block of the score. A linear equation graphs as a straight line and has the form y = mx + b (slope m, y-intercept b).

Solving a single equation

Isolate the variable by doing the same operation to both sides:

5x − 3 = 2x + 9

3x = 12 → x = 4

Systems of equations

Two lines, solved by substitution or elimination. The solution is where they intersect.

y = 2x + 1 and y = −x + 7

2x + 1 = −x + 7 → 3x = 6 → x = 2, y = 5 → (2, 5)

Number of solutions: one (lines cross), none (parallel, same slope, different intercept), or infinite (same line).

Worked example

3x + 2y = 12 and x − y = 1. Solve.

From the second equation, x = y + 1. Substitute: 3(y + 1) + 2y = 12 → 5y = 9 → y = 1.8, x = 2.8.

SAT strategy

The Digital SAT gives you a built-in Desmos graphing calculator. For systems or "number of solutions" questions, graph both equations and read the intersection — it is often faster and less error-prone than algebra. Type each equation and tap the intersection point.