System of Equations Solver (2×2)
Two equations, two unknowns, one click. We'll solve for x and y so you don't have to.
📖 How it works & FAQWhat a 2x2 system solver does
A 2x2 system is two linear equations that share the same two unknowns, usually written as a1x + b1y = c1 and a2x + b2y = c2. Solving it means finding the single pair of values for x and y that makes both equations true at once. Geometrically, each equation is a straight line, and the solution is the point where the two lines cross.
When you'd reach for this
It comes up constantly in algebra homework, but also in the real world: splitting a fixed budget between two items, mixing two solutions to hit a target concentration, finding where supply meets demand, or working out two unknown prices from two receipts. Any time you have two facts and two things you don't know, this is the tool.
A worked example
Say 2x + 3y = 12 and x − y = 1. The solver uses elimination (Cramer's rule under the hood): it computes the determinant a1b2 − a2b1, here (2)(−1) − (1)(3) = −5, then divides through to get x = 3, y = 2. Plug those back in and both equations check out.
How to use it
- Enter the six numbers — the coefficients of x and y and the constant — for each of your two equations. Move every variable to the left side and the constant to the right first.
- Use negative signs and decimals or fractions exactly as they appear; an empty coefficient counts as zero.
- Press Solve to read off the exact x and y values, along with the crossing point.
- If you get "no solution" or "infinitely many," double-check your signs — that result means the lines are parallel or identical.
FAQ
- Why does it say there's no unique solution?
- When the determinant is zero, the two lines never cross once. Parallel lines give no solution; the same line written twice gives infinitely many.
- Can I enter fractions or decimals?
- Yes. Type coefficients like 0.5, −3, or 7 directly. The math runs the same way regardless of format.
- Does anything get uploaded?
- No. The calculation happens entirely in your browser, so your numbers never leave your device.
- Is it free?
- Completely free and ad-supported, with no sign-up.