Overview
A system of three equations in three variables represents three planes in space. A unique solution is the point where all three planes meet. This calculator finds it with Cramer's rule or detects when no unique solution exists.
Formula
a₁x + b₁y + c₁z = d₁, and so on. x = Dₓ/D, y = D_y/D and z = D_z/D when D ≠ 0.
How it works
Write the three equations
Make sure each equation is in the form ax + by + cz = d.
Compute the coefficient determinant
D = |A|. If it is zero, there is no unique solution.
Replace the columns
Compute Dₓ, D_y and D_z by replacing each variable column with the constants column.
Divide and verify
Divide each determinant by D, then substitute into all three equations.
Worked examples
Example: x + y + z = 6
With 2x − y + z = 3 and x + 2y − z = 2, the solution is x = 1, y = 2 and z = 3.
No unique solution: x + y + z = 1
With 2x + 2y + 2z = 2 and 3x + 3y + 3z = 4, the equations contradict each other and there is no solution.
Frequently asked questions
What does a zero determinant mean here?
There is no unique solution; the planes may be parallel, intersect along a line, or coincide.
Does the method work for any coefficients?
Yes, though manual computation becomes long. The calculator handles the 3×3 determinants.
How do I know the solution is correct?
Substitute x, y and z into all three equations; each right side should match.
This calculator is for educational and informational purposes only and is not a substitute for professional advice.