Systems of Two Linear Equations
A system's solution is wherever two lines meet — visually on a graph, or algebraically through substitution and elimination. All three are the same idea seen three ways.
Two linear equations, graphed together, can relate in exactly three ways. Different slopes means the lines cross exactly once — one solution, marked in amber above. Same slope, different intercept means the lines are parallel and never meet — no solution. Same slope and same intercept means they’re the same line — infinitely many solutions.
You won’t always get to graph it on the SAT, though — the other two tabs walk through the two algebraic methods that get you to the same answer without a picture.
A common trick question: two equations that look different but simplify to the same line (infinitely many solutions), or the same slope in disguise (no solution). Rewriting both equations in the same form before comparing slopes — or before choosing substitution vs. elimination — avoids that trap.