3 Variable System of Equations Calculator (Cramer's Rule)
By Santaji GadepatilinUpdated November 6, 2026
Three linear equations in three unknowns describe three planes, and when they meet at a single point the system has one solution. This 3 variable system of equations calculator finds that point with Cramer's rule, showing the coefficient determinant and the three replacement determinants so you can follow the working.
How to use the 3 variable system of equations calculator
Write each equation as a·x + b·y + c·z = d and enter its four numbers. Use 0 for a missing variable. The calculator returns D, Dx, Dy, Dz and the solution; if D = 0 there is no unique solution.
Cramer’s rule for three equations
D = a₁(b₂c₃ − c₂b₃) − b₁(a₂c₃ − c₂a₃) + c₁(a₂b₃ − b₂a₃)
x = Dx/D · y = Dy/D · z = Dz/D
Dx, Dy and Dz are the same determinant with the x, y or z column replaced by the right-hand sides d.
Worked example
Solve x + y + z = 6, 2x − y + 3z = 9 and x + 2y − z = 2:
- D = 1(1 − 6) − 1(−2 − 3) + 1(4 + 1) = 5
- Dx = 5, Dy = 10, Dz = 15
- x = 5/5 = 1, y = 10/5 = 2, z = 15/5 = 3
- Check: 1 + 2 + 3 = 6 ✓, 2 − 2 + 9 = 9 ✓, 1 + 4 − 3 = 2 ✓
Solving by elimination
- Use equation 1 to eliminate x from equations 2 and 3.
- Solve the resulting two equations in y and z (see the two-variable system of equations calculator).
- Substitute y and z back into equation 1 to find x.
Frequently asked questions
How do you solve 3 equations with 3 variables?
Use elimination to reduce to two equations in two unknowns, or Cramer's rule — x = Dx/D, y = Dy/D, z = Dz/D, where each D is a 3×3 determinant.
How is a 3×3 determinant calculated?
By cofactor expansion along the first row, D = a₁(b₂c₃ − c₂b₃) − b₁(a₂c₃ − c₂a₃) + c₁(a₂b₃ − b₂a₃).
What if the determinant is zero?
There is no unique solution — the planes are parallel or meet in a line, giving no solutions or infinitely many. Gaussian elimination tells you which.
Is Cramer's rule efficient?
For 2×2 and 3×3 systems it is quick and clear. For larger systems Gaussian elimination or matrix methods are far more efficient.