Simultaneous Equations Calculator
Solve two linear equations in x and y with Cramer's rule, shown step by step, and see when there's no solution or infinitely many.
x
3.2
- y
- 2.2
The lines meet at x = 3.2, y = 2.2.
How it's worked out
- Equations2x + 3y = 13; 1x − 1y = 1
- Determinant
2 × -1 − 1 × 3-5 - x (Cramer's rule)
(13 × -1 − 1 × 3) ÷ -53.2 - y
(2 × 1 − 1 × 13) ÷ -52.2
Cramer's rule
D = a₁b₂ − a₂b₁ · x = (c₁b₂ − c₂b₁) ÷ D · y = (a₁c₂ − a₂c₁) ÷ D
Worked example
For 2x + 3y = 13 and x − y = 1: D = 2 × (−1) − 1 × 3 = −5, x = (13 × (−1) − 1 × 3) ÷ (−5) = 3.2 and y = (2 × 1 − 1 × 13) ÷ (−5) = 2.2.
Questions people ask
How do I solve simultaneous equations?
By elimination, substitution or Cramer's rule. With Cramer's rule, x = (c₁b₂ − c₂b₁) ÷ (a₁b₂ − a₂b₁) and y = (a₁c₂ − a₂c₁) ÷ (a₁b₂ − a₂b₁).
When is there no solution?
When the two lines are parallel: the determinant a₁b₂ − a₂b₁ is 0 but the equations aren't the same line.
When are there infinitely many solutions?
When both equations describe the same line, such as x + y = 2 and 2x + 2y = 4.
About this calculator
Formula and rules last checked 25 October 2026Written and tested by the CalculatorDex teamReport a problem