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System of Linear Equations Solver (2 Equations)

Solve two linear equations in two unknowns (a1x + b1y = c1, a2x + b2y = c2) with Cramer's rule, including the parallel-lines edge cases.

x
2
y
1

How it works

Cramer's rule solves a1x + b1y = c1 and a2x + b2y = c2 using determinants. The main determinant D = a1b2 − a2b1 comes from the coefficient matrix; when it's nonzero, the two lines cross at exactly one point: x = (c1b2 − c2b1) ÷ D and y = (a1c2 − a2c1) ÷ D.

When D = 0, the two lines have the same slope, so they never cross at a single point — they're either the exact same line (infinitely many solutions, when the constants are proportional too) or two distinct parallel lines that never meet (no solution). This calculator checks for both cases explicitly rather than dividing by zero.

FAQ

What does it mean if the calculator says there are infinitely many solutions?

It means both equations describe the exact same line, just written differently (e.g. one is a multiple of the other, like 2x+4y=6 and x+2y=3) — every point on that line satisfies both equations, so there's no single unique answer.

Can this solve three equations in three unknowns?

No — this solves exactly two equations in two unknowns (x and y). A 3-variable system needs a 3x3 determinant and a different input layout, which isn't what this calculator is built for.

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