Systems of Equations Solver (3 Variables)
Solve a system of 3 linear equations with 3 unknowns.
Enter coefficients and get the solution for x, y, and z.
Solution
A system of 3 linear equations with 3 unknowns looks like:
a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
Solution method: This calculator uses Cramer’s Rule with determinants.
The determinant of the coefficient matrix is:
D = a₁(b₂c₃ - b₃c₂) - b₁(a₂c₃ - a₃c₂) + c₁(a₂b₃ - a₃b₂)
Then: x = Dx/D, y = Dy/D, z = Dz/D
When the determinant is zero, the system either has no solution (inconsistent) or infinitely many solutions (dependent).
Example:
2x + y - z = 8
-3x - y + 2z = -11
-2x + y + 2z = -3
Solution: x = 2, y = 3, z = -1