System of Equations Calculator
Enter the six coefficients of two linear equations and get the unique x and y where the lines meet — solved instantly with Cramer's rule.
x and y at once
Enter a₁, b₁, c₁, a₂, b₂ and c₂ for the two equations and the calculator returns the unique solution (x, y) plus the determinant D.
Standard form
Rearrange each equation into a·x + b·y = c before you start, so every coefficient lines up in the right box.
What is a 2×2 system of equations?
Two lines, one meeting point
A system of equations calculator finds the pair of values (x, y) that make two linear equations true at the same time. Each equation, written as a·x + b·y = c, is a straight line on a graph, and solving the system means locating the single point where those two lines cross. This tool takes the six coefficients — a₁, b₁, c₁ from the first equation and a₂, b₂, c₂ from the second — and applies Cramer's rule to return x and y exactly, along with the determinant that decides whether a unique answer even exists.
Type the six coefficients and read off the unique x and y where the two lines intersect — no algebra by hand required.
Cramer's rule expresses each unknown as a ratio of determinants. First compute the coefficient determinant D, then divide two further determinants by it.
D = a₁b₂ − a₂b₁, x = (c₁b₂ − c₂b₁)/D, y = (a₁c₂ − a₂c₁)/DTake the worked example above: the equations are 2x + y = 5 and x − y = 1. The determinant is D = 2·(−1) − 1·1 = −3, which is not zero, so a unique solution exists. Then x = (5·(−1) − 1·1) / −3 = (−6)/−3 = 2, and y = (2·1 − 1·5) / −3 = (−3)/−3 = 1. Substituting back confirms it: 2·2 + 1 = 5 and 2 − 1 = 1, so (x, y) = (2, 1) is correct.
Cramer's rule is exact, but a couple of points are worth keeping in mind.
No unique solution when D = 0
When the determinant D = a₁b₂ − a₂b₁ equals zero, the two lines are parallel (no solution) or identical (infinitely many solutions), and there is no single (x, y) to report — so the calculator returns no result. Make sure each equation is in standard a·x + b·y = c form before entering the coefficients, and remember that any coefficient may legitimately be negative or zero; only a zero determinant blocks a unique answer.