Micro Tools

Linear System 2x2 Solver

Solve a 2x2 linear system (two equations, two unknowns) with Cramer's rule, step by step.

Linear System 2x2

Enter both equations. Results update live.

a1x + b1y = c1

a2x + b2y = c2

Result

Enter numbers to see the result.

Determinant

Formula: D = a1b2 - a2b1. x = (c1b2 - c2b1) / D, y = (a1c2 - a2c1) / D.

Cramer's rule in action

Cramer's rule replaces one column of the coefficient matrix with the constants and divides determinants. The main determinant D = a1b2 - a2b1 decides everything: if it is non-zero there is exactly one solution, and the tool computes x and y from it directly.

When there is no unique solution

A zero determinant means the two lines are parallel (no solution at all) or the same line (infinitely many solutions). The tool reports this instead of dividing by zero, so you always know what the system is telling you.

Linear System 2x2 FAQs

How do I solve 2x + 3y = 7 and x - y = 1?

Enter the coefficients: a1=2, b1=3, c1=7, a2=1, b2=-1, c2=1. The solution is x = 2, y = 1. You can verify: 2(2) + 3(1) = 7.

What is Cramer's rule?

It solves a linear system using determinants: x equals the determinant with the constants swapped into the x column, divided by the main determinant. It works whenever the main determinant is not zero.

What does no unique solution mean?

Either the equations describe parallel lines that never meet, or they describe the same line, giving infinitely many solutions. In both cases the determinant is zero.

Can I use decimals or fractions?

Yes. Type decimals directly; for fractions like 1/3 enter 0.3333. The solver works with any real coefficients.

Does the solver send my equations anywhere?

No. The computation happens in your browser with plain JavaScript.

Linear System 2x2

Cramer's rule in action

When there is no unique solution

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Linear System 2x2 FAQs