Simultaneous Equation Solver
Solve a system of two linear equations in two unknowns using Cramer's rule.
Inputs
x
2.200000
y
1.200000
Determinant
-5.000000
Step by step
Values used
a₁ (eq1 coeff of x) = 2; b₁ (eq1 coeff of y) = 3; c₁ (eq1 constant) = 8; a₂ (eq2 coeff of x) = 1; b₂ (eq2 coeff of y) = -1; c₂ (eq2 constant) = 1
Determinant
det = a₁·b₂ − a₂·b₁
Solution
x = (c₁·b₂ − c₂·b₁) / det, y = (a₁·c₂ − a₂·c₁) / det
x
= 2.200000
y
= 1.200000
Determinant
= -5.000000
How it works
Cramer's rule solves a 2×2 system by computing the determinant of the coefficient matrix and substituting columns with the constants vector. If det = 0, the lines are parallel (no solution) or coincident (infinite solutions).
Formulas
Determinant
det = a₁·b₂ − a₂·b₁
- a1
- Eq1 x-coefficient
- b1
- Eq1 y-coefficient
- a2
- Eq2 x-coefficient
- b2
- Eq2 y-coefficient
Solution
x = (c₁·b₂ − c₂·b₁) / det, y = (a₁·c₂ − a₂·c₁) / det
- c1
- Eq1 constant
- c2
- Eq2 constant
Frequently Asked Questions
What happens when the determinant is zero?
The two lines are either parallel (no solution) or identical (infinitely many solutions). This calculator reports an error in that case.
Can I solve 3×3 systems?
This calculator handles 2×2 systems. For larger systems, use the matrix calculator or Gaussian elimination.