What This Calculator Does

This tool solves a system of three linear equations in three unknowns (x, y, z). You provide the coefficients of the equations in standard form:

$a_1x + b_1y + c_1z = d_1$
$a_2x + b_2y + c_2z = d_2$
$a_3x + b_3y + c_3z = d_3$

The calculator returns the unique values of x, y and z, along with the coefficient determinant $\det(A)$. It works for any real coefficients, including negatives, fractions and decimals.

How to Use It

Each row corresponds to one equation. Type the coefficient in front of x (a), y (b) and z (c), then the constant on the right-hand side (d). Move every variable to the left and the constant to the right before entering values. If your equation is written as 5 = 2x − y, rewrite it as 2x − y + 0z = 5.

The Formula Explained

The solution uses Cramer's rule. First we compute the determinant of the coefficient matrix A. Then for each variable we replace the matching column of A with the constants column d and take that determinant. Dividing gives the variable:

$$x = \dfrac{\det(A_x)}{\det(A)},\quad y = \dfrac{\det(A_y)}{\det(A)},\quad z = \dfrac{\det(A_z)}{\det(A)}$$

If $\det(A) = 0$, Cramer's rule fails — the system has no unique solution (either no solution or infinitely many), and the calculator flags this.

Rule of Sarrus diagonals for computing a 3x3 determinant — The 3×3 determinant via Sarrus: sum the down-diagonals and subtract the up-diagonals.

Coefficient matrix A and three matrices with replaced columns for Cramer's rule — Cramer's rule: each variable equals $\det(A_i)$ over $\det(A)$, where $A_i$ replaces one column with the constants.

Worked Example

Solve $2x + y - z = 8$, $-3x - y + 2z = -11$, $-2x + y + 2z = -3$.

$\det(A) = -1$. Applying Cramer's rule gives $\det(A_x) = -2$, $\det(A_y) = -3$, $\det(A_z) = 1$, so $x = 2$, $y = 3$, $z = -1$. You can verify:

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

Interpreting Your Result

Each equation in a 3×3 system describes a plane in three-dimensional space. The solution is wherever all three planes meet, and the value of the determinant $D=\det(A)$ tells you which of three cases you are in.

$D\neq0$: unique solution

When the coefficient determinant is nonzero, the three planes intersect at exactly one point. Cramer's rule returns a single $(x,y,z)$, and that ordered triple is the only set of values satisfying all three equations at once. This is a consistent, independent system. The output $x=D_x/D$, $y=D_y/D$, $z=D_z/D$ is exact (within rounding) and can be checked by substituting back into the original equations.

$D=0$: no unique solution

When $D=0$ the matrix is singular and Cramer's rule cannot divide. Two sub-cases exist:

Inconsistent — no solution. The planes have no common point (for example, two or more are parallel, or they form a triangular prism arrangement where no single point lies on all three). The system has zero solutions.
Dependent — infinitely many solutions. The planes share a whole line (or coincide). Here the equations are not independent, and there is an infinite family of $(x,y,z)$ triples, usually described with a free parameter.

The determinant alone cannot distinguish these two; you must inspect the equations (e.g., via row reduction) to see whether they are contradictory or redundant.

Reading the x, y, z output

The three returned numbers are the coordinates that make every equation true. A value can be negative, zero, or fractional. If the calculator reports $D=0$, treat the answer with caution and re-examine the system rather than trusting a divided result.

Definitions & Glossary

Coefficient matrix $A$: The 3×3 array of the numbers multiplying $x, y, z$ on the left side of each equation: rows are the equations, columns correspond to $x$, $y$, and $z$.
Constants vector $d$: The column $(d_1, d_2, d_3)$ of right-hand-side values that the equations equal.
Determinant $\det(A)$ (also $D$): A single scalar computed from a square matrix that measures whether the matrix is invertible. $\det(A)\neq0$ means a unique solution exists.
Cramer's rule: A method that solves a square linear system by writing each variable as a ratio of determinants: $x=D_x/D$, $y=D_y/D$, $z=D_z/D$, where $D_x, D_y, D_z$ come from replacing the matching column with $d$.
Sarrus' rule: A shortcut for the determinant of a 3×3 matrix: sum the three diagonals running top-left to bottom-right and subtract the three diagonals running top-right to bottom-left.
Singular matrix: A square matrix whose determinant is $0$; it has no inverse, so Cramer's rule yields no unique solution.
Unique solution: Exactly one $(x,y,z)$ satisfies the system; occurs when $D\neq0$.
Consistent system: A system that has at least one solution (one or infinitely many).
Dependent system: A consistent system with infinitely many solutions because the equations are not all independent.
Inconsistent system: A system with no solution at all; its equations contradict one another.
$a, b, c, d$ per row: Within row $i$, $a_i$ is the $x$-coefficient, $b_i$ the $y$-coefficient, $c_i$ the $z$-coefficient, and $d_i$ the constant on the right-hand side.

FAQ

What if det(A) is zero? The three planes do not meet at a single point, so there is no unique (x, y, z). The system is either inconsistent or dependent.

Can I use decimals or fractions? Yes — enter decimals directly (use 0.5 instead of 1/2).

Is Cramer's rule accurate? For a 3×3 system it is exact and stable for typical inputs. Very large or near-singular systems may show small rounding in the last decimals.

3×3 System of Equations Calculator

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