Simultaneous Equations

GRE Quantitative · Axiom Academy

Two roads, one destination — substitution and elimination both drive straight at the single point where a system's equations agree. 1. A System Is Two Lines Looking for Their Crossing Take 2x + y = 5 and x - y = 1 . Each is a line. A pair (x,y) solves both equations exactly when it lies on both lines — which happens at exactly one place: where they cross. 2. Substitution: Solve One, Feed It Into the Other Solve one equation for one variable, then substitute that expression into the other equation — collapsing two unknowns down to one. . Then x = y+1 = 2 . Same point: (2,1) . 3. Elimination: Combine the Equations Themselves Instead of solving for a variable first, add or subtract the whole equations so one variable cancels outright. . Substitute back into either original equation: . Same point: (2,1) — again. 4. When There's No Single Crossing Point Both methods assume the lines cross exactly once. Two other things can happen: the lines never meet, or they're really the same line. x+y=3 and x+y=5 : same slope, different intercept. Elimination gives 0=2 — a false statement, so there's no (x,y) that works. x+y=3 and 2x+2y=6 : the second is just the first doubled. Every point on the line works — infinitely many solutions. Substitution and elimination are two paths to the same intersection point — and when there isn't a single point, the algebra tells you that too. Scroll up to revisit any step.

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