System of Linear Equations Calculator

Solve a1x + b1y = c1 and a2x + b2y = c2 for x and y.

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How to Use the System of Linear Equations Calculator

  1. Enter the coefficients for the first equation (a1x + b1y = c1).
  2. Enter the coefficients for the second equation (a2x + b2y = c2).
  3. Click Calculate to solve for x and y.

Formula

x = (c1b2 − c2b1) ÷ (a1b2 − a2b1); y = (a1c2 − a2c1) ÷ (a1b2 − a2b1)

Worked Example

For 2x + 3y = 8 and x - y = 1, the solution is x = 2.2 and y = 1.2 — the point where both lines cross.

About This Calculator

This solves any 2x2 linear system using Cramer's Rule. If the two lines are parallel (determinant = 0), there is no single unique solution.

Frequently Asked Questions

What does this calculator solve?

It solves a system of two or more linear equations simultaneously for the unknown variables.

What methods are commonly used?

Substitution, elimination, and matrix-based methods like Cramer's rule or Gaussian elimination.

What does no solution mean?

The equations represent parallel lines that never intersect, so no values satisfy both.

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