Quick start
- Write each equation as ax + by + cz = d, moving everything else to the right side.
- Enter the four numbers for each row. Use 0 when a variable is missing, so x + z = 5 is 1, 0, 1, 5.
- Read D, Dx, Dy and Dz, then the values of x, y and z.
How the 3×3 determinant is found
Expand along the top row: take each entry, multiply it by the 2×2 determinant left when you cover its row and column, and alternate the signs +, −, +. Cramer's rule then works exactly as in the 2×2 case: replace one column with the constants, take the determinant, and divide by D.
Example
For x + y + z = 6, 2x − y + z = 3 and x + 2y − z = 2, the main determinant is 7. The solution is x = 1, y = 2 and z = 3, which you can check by substituting into all three equations.
No unique answer?
If D = 0, the three planes do not meet at one point. They may share a line, overlap completely, or never meet at all.
Common questions
Why use determinants instead of elimination?
Both work. Determinants give a fixed recipe that is easy to check, while elimination is often quicker by hand.
Can it handle decimals and negatives?
Yes. Any real coefficients work.
What does D = 0 mean in real terms?
The equations are not independent. One can be built from the others, or they contradict each other.