A system of two linear equations asks for pairs (x, y) satisfying both equations simultaneously. Each nondegenerate equation defines a line.
a1 and b1 are the first equation's coefficients and c1 its right-hand side; a2, b2 and c2 belong to the second equation. Zero coefficients are allowed.
Intersecting lines have one solution, coincident lines have infinitely many and distinct parallel lines have none. Degenerate equations must be examined too: 0 = 0 adds no restriction, while 0 = c with c ≠ 0 is impossible.
Enter all six coefficients in the order a1, b1, c1, a2, b2, c2 and solve. Use zero explicitly for absent terms. Editing any field clears the old solution and steps.
For x + y = 3 and x − y = 1, subtraction gives −2y = −2. Therefore y = 1 and x = 2; the solution is (2, 1).
An equation ax = c with nonzero a is vertical. Coincident lines overlap. An identity describes the entire plane, not a line; a contradiction has no geometric points. The chart shows the nondegenerate equations even when the system is incompatible.
The root, intersection or an intercept may lie outside the graph window. For coordinates beyond 10⁶, only the window [−10, 10] is shown; the exact result remains valid.
Systems model two simultaneous constraints, such as two prices inferred from purchases or the meeting of two constant-speed trajectories.