Simultaneous Equations (3 Variables) Calculator

Work out simultaneous equations (3 variables) instantly with clear inputs, formula shown and shareable results.

About Simultaneous Equations (3 Variables) Calculator

Cramer's rule extends to three unknowns: each variable equals the determinant of the coefficient matrix with its column replaced by the right-hand side, divided by the coefficient determinant.

Formula

Cramer's rule (3×3)

x = det(Aₓ)/det(A), y = det(A_y)/det(A), z = det(A_z)/det(A)

Worked example

With the default values below, Simultaneous Equations (3 Variables) Calculator returns x of 2.00000000. Change any input above and every figure here is recalculated in your browser.

Inputs used

First equation a, b, c, d (for ax + by + cz = d): 2, 1, -1, 8
Second equation a, b, c, d: -3, -1, 2, -11
Third equation a, b, c, d: -2, 1, 2, -3

Results

x: 2.00000000
y: 3.00000000
z: -1.00000000
Coefficient determinant: -1.00000000

Step by step

1Values used
2Cramer's rule (3×3)
3x= 2.00000000
4y= 3.00000000
5z= -1.00000000
6Coefficient determinant= -1.00000000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Solve the sample system.

2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3 gives x = 2, y = 3, z = −1.

What if the determinant is zero?

The planes either meet in a line, share a plane, or have no common point. Row reduction is needed to tell which.