Last updated: 2026-07-01

What this calculator does

This calculator solves a system of three linear equations in three unknowns, x, y, and z:

$$a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3$$

Given the twelve coefficients and constants, it applies Cramer's rule to return the unique solution (x, y, z), or reports that the system has no unique solution when the coefficient matrix is singular.

Cramer's rule for a 3×3 system

Cramer's rule expresses each unknown as a ratio of determinants. First compute the determinant of the coefficient matrix:

$$D = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = a_1(b_2 c_3 - c_2 b_3) - b_1(a_2 c_3 - c_2 a_3) + c_1(a_2 b_3 - b_2 a_3)$$

When D ≠ 0, the unique solution is:

$$x = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}$$

Each numerator determinant is the coefficient determinant with one column replaced by the constants column (d₁, d₂, d₃):

$$D_x = \begin{vmatrix} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \end{vmatrix}, \quad D_y = \begin{vmatrix} a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3 \end{vmatrix}, \quad D_z = \begin{vmatrix} a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3 \end{vmatrix}$$

This comes straight from linear algebra: for a non-singular matrix A, the solution of Av = d is v = A⁻¹d, and Cramer's rule is what that inverse looks like written out determinant by determinant.

Geometric meaning

Each equation describes a plane in three-dimensional space. Solving the system means finding the point where all three planes meet.

• D ≠ 0 (unique solution): The three planes intersect at exactly one point. The coefficient matrix is invertible.
• D = 0 (no unique solution): The planes are arranged so they do not meet at a single point. They may share a common line (infinitely many solutions) or have no common point at all (no solution). Cramer's rule cannot return a single answer in this case.

Worked example

Solve the default system:

$$2x + y - z = 8 \\ -3x - y + 2z = -11 \\ -2x + y + 2z = -3$$

Step 1 — determinant of the coefficient matrix:

$$D = 2\big((-1)(2) - (2)(1)\big) - 1\big((-3)(2) - (2)(-2)\big) + (-1)\big((-3)(1) - (-1)(-2)\big)$$ $$D = 2(-4) - 1(-2) + (-1)(-5) = -8 + 2 + 5 = -1$$

D ≠ 0, so there is a unique solution.

Step 2 — solve for x (replace the x-column with the constants):

$$D_x = \begin{vmatrix} 8 & 1 & -1 \\ -11 & -1 & 2 \\ -3 & 1 & 2 \end{vmatrix} = -2 \quad\Rightarrow\quad x = \frac{-2}{-1} = 2$$

Step 3 — solve for y (replace the y-column with the constants):

$$D_y = \begin{vmatrix} 2 & 8 & -1 \\ -3 & -11 & 2 \\ -2 & -3 & 2 \end{vmatrix} = -3 \quad\Rightarrow\quad y = \frac{-3}{-1} = 3$$

Step 4 — solve for z (replace the z-column with the constants):

$$D_z = \begin{vmatrix} 2 & 1 & 8 \\ -3 & -1 & -11 \\ -2 & 1 & -3 \end{vmatrix} = 1 \quad\Rightarrow\quad z = \frac{1}{-1} = -1$$

Verification:

$$2(2) + 3 - (-1) = 8 \checkmark \quad -3(2) - 3 + 2(-1) = -11 \checkmark \quad -2(2) + 3 + 2(-1) = -3 \checkmark$$

When the determinant is zero

If D = 0, the three rows of the coefficient matrix are linearly dependent — one equation is a combination of the others. Two outcomes are possible:

• Infinitely many solutions: the constants are consistent with the dependency, so the planes share a whole line of common points.
• No solution: the constants contradict the dependency, so the planes never meet at one point.

Cramer's rule applies only when D ≠ 0, which is why the calculator reports "no unique solution" the moment D = 0. To classify the degenerate case precisely, reduce the augmented matrix with Gaussian elimination and compare the rank of the coefficient matrix with the rank of the augmented matrix.

Applications

Three-variable linear systems appear throughout algebra and precalculus coursework, in circuit analysis (Kirchhoff's laws across three loops), in chemistry when balancing reactions, and in any setting where three linear conditions must hold at once. Cramer's rule gives a direct closed-form answer, which makes it a convenient way to check a small system or to express a solution symbolically.

Frequently Asked Questions (FAQ)

What does the determinant tell me?

The determinant D of the coefficient matrix decides whether the system has one clean answer. When D ≠ 0 the three planes meet at exactly one point, so x, y, and z are uniquely determined. When D = 0 the planes are arranged so that they share a whole line or no common point at all, and Cramer's rule cannot return a single solution.

How is Cramer's rule different from elimination?

Gaussian elimination and substitution reach the same answer by step-by-step row operations. Cramer's rule instead writes each unknown directly as a ratio of determinants: x = Dₓ/D, y = D_y/D, z = D_z/D. It is convenient for a closed-form expression and for systems you want to evaluate symbolically, though elimination is usually faster for larger systems.

Why does my system show "no unique solution"?

That message appears whenever D = 0. The three equations are then linearly dependent: at least one is a combination of the others. Depending on the constants, the system either has infinitely many solutions (the planes share a line) or none (the planes never meet at a common point). Cramer's rule only applies when D ≠ 0, so it stops here.

How can I verify the solution?

Substitute the returned x, y, and z back into all three original equations. Each left-hand side should equal its right-hand side (d₁, d₂, d₃). Because the solution is computed from exact determinant ratios, small rounding in the displayed decimals is normal; the underlying values satisfy the equations to full precision.