What are simultaneous equations?
Equations that share the same unknowns and must be true together.
Simultaneous Equations Calculator
Solve two or three linear equations in as many unknowns using elimination/determinants.
Choose system size
Enter coefficients for ax+by(+cz)=d form.
About this calculator
How to use Simultaneous Equations Calculator, plus the formulas and logic behind the results.
Tool description
A linear system has a unique solution when its coefficient determinant is nonzero. 2×2 uses Cramer-style formulas; 3×3 uses determinant expansion.
How to use
- Select 2×2 or 3×3.
- Enter all coefficients and right-hand sides.
- For 2×2, z-related fields are ignored.
- Select Calculate to view the solution or an error if singular.
Formulas & reference
2×2: x = (d1 b2 − d2 b1)/Δ, y = (a1 d2 − a2 d1)/ΔΔ = a1 b2 − a2 b13×3: Cramer’s rule with det(A), det(A_x), det(A_y), det(A_z)
Tips
- Unique solution ⇔ nonzero determinant
- Check substitution into each equation
- 2×2 sample: 2x+y=8, x−y=1 → check by substitution
- Floating-point rounding may appear in decimals
Calculation logic
Builds the coefficient matrix, checks the determinant, and applies Cramer’s rule for unique solutions.
Singular systems (Δ = 0) may have none or infinitely many solutions; this tool reports no unique solution.
Simultaneous Equations Calculator FAQ
Common questions about this calculator and how to use the results.
What if there is no unique solution?
The determinant is zero—lines/planes are parallel or dependent.
Do I need z for 2×2?
No—those fields are ignored in 2×2 mode.
Is Gaussian elimination used?
Results match elimination; implementation uses determinants/Cramer.
Can coefficients be decimals?
Yes.
How do I verify?
Substitute the printed x,y(,z) back into each equation.