Simultaneous Equation Solver
Solve a system of two linear equations in two unknowns using Cramer's rule.
About Simultaneous Equation Solver
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
Worked example
With the default values below, Simultaneous Equation Solver returns x of 2.200000. Change any input above and every figure here is recalculated in your browser.
Inputs 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
Results
- x
- 2.200000
- y
- 1.200000
- Determinant
- -5.000000
Step by step
- 1Values used
- 2Determinant
- 3Solution
- 4x= 2.200000
- 5y= 1.200000
- 6Determinant= -5.000000
Assumptions and limits
- The result is derived only from the 6 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.