Simultaneous equations solver
Solve two linear equations with two unknowns, x and y, using Cramer's rule.
Enter your values and press Calculate. Your inputs stay in the address, so you can switch language or share the link.
Formula
a₁x + b₁y = c₁, a₂x + b₂y = c₂ → x = (c₁b₂ − c₂b₁) ÷ D, y = (a₁c₂ − a₂c₁) ÷ D, D = a₁b₂ − a₂b₁
How it works
Each equation has the form a·x + b·y = c. Cramer's rule divides a determinant built from the c-values by the determinant of the coefficients to find each unknown directly.
Worked example
The same calculation with the default values:
- a₁ (equation 1) 2
- b₁ (equation 1) 1
- c₁ (equation 1) 8
- a₂ (equation 2) 1
- b₂ (equation 2) -1
- c₂ (equation 2) 1
- D = 2 × -1 − 1 × 1 = -3
- x = (8 × -1 − 1 × 1) ÷ -3 = 3
- y = (2 × 1 − 1 × 8) ÷ -3 = 2
Result: x: 3 · y: 2
If the two equations describe the same line there are infinitely many solutions; if they are parallel but different, there is none.
Common mistakes
- Forgetting to check the determinant D before trusting the result - when D = 0, the two lines are parallel (no solution) or identical (infinite solutions), and the x/y formulas become meaningless division by zero.
- Mixing up which coefficient belongs to which equation when entering values - a₁,b₁,c₁ must all come from the first equation and a₂,b₂,c₂ from the second, not mixed between them.
- Assuming a solution always exists - two real-world constraints can be contradictory (no solution) or redundant (infinitely many), not just a single intersection point.
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