System of Equations Solver

Enter the coefficients of a₁x + b₁y = c₁ and a₂x + b₂y = c₂.

Result

Enter values and press Calculate to see the result.

Formula

x=c1b2−c2b1a1b2−a2b1,  y=a1c2−a2c1a1b2−a2b1x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1},\; y = \frac{a_1 c_2 - a_2 c_1}{a_1 b_2 - a_2 b_1}

Cramer’s rule, valid when the determinant a₁b₂ − a₂b₁ is not zero.

How to use this calculator

  1. Write both equations in the form ax + by = c.
  2. Enter all six numbers and press Calculate.

Worked example

Solve 2x + 3y = 8 and x − y = −1

D = 2·(−1) − 1·3 = −5.

x = (8·(−1) − (−1)·3)/(−5) = 1 and y = (2·(−1) − 1·8)/(−5) = 2.

Frequently asked questions

What if the determinant is zero?

The lines are parallel (no solution) or the same line (infinitely many solutions). Sumzo tells you which.

Can I solve three equations?

Not yet in this calculator. Sumzo AI can help with larger systems.

Learn more