3x3 System Calculator

Solve three equations in three unknowns. Each equation is a plane, and a unique solution is the single point where all three meet.

Solution

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Solution—

Determinant of the system—

Type of system—

Determinant of x—

Determinant of y—

Determinant of z—

How it was solved

The formula

x = Δx ÷ Δ · y = Δy ÷ Δ · z = Δz ÷ Δ

Why it works

With three unknowns each equation describes a plane in space. A non-zero determinant means the three planes meet at exactly one point. A zero determinant means they do not: they may share a line, or form a triangular prism and never all meet.

How to solve it by hand

Write the coefficients of each equation as a row: x, y, z, result
Work out the determinant of the 3×3 coefficient matrix
Replace each column in turn with the constants for Δx, Δy and Δz
Divide each by the determinant

What is worth knowing

The three-plane picture is the clearest way to understand why a zero determinant is ambiguous. Three planes can fail to meet at a point in several ways: all three parallel, two parallel and one crossing, or three planes meeting pairwise in three parallel lines like the sides of a triangular prism. The determinant tells you the failure happened but not which kind, which is why the full analysis uses the rank of the augmented matrix. For hand calculation, the determinant is usually enough to know whether to keep going.

Frequently asked questions

What does the determinant tell you here?

Whether the three planes meet at a single point. Non-zero means yes; zero means they do not, though it does not say why.

Why does the answer sometimes look like 0.9999999?

Floating-point arithmetic. The exact answer is 1, and the tiny difference comes from how computers store decimals.

Can I solve it by elimination instead?

Yes, and for larger systems you should. Gaussian elimination scales far better than determinants.

What if I have four unknowns?

The same principles apply, but Cramer's rule becomes impractical. Use elimination or matrix methods.