Three-Variable System of Equations Calculator
Solve three linear equations from free-form input with x, y, and z, including parentheses, fractions, Cramer's Rule steps, rank checks, and substitution verification.
Three equations with three unknowns: enter equations directly, then check the parsed system, determinant test, and x, y, z result.
Enter three equations directly
Use this calculator when you have three full linear equations in x, y, and z rather than a prepared coefficient matrix. It parses equations with parentheses, fractions, decimals, and variables on either side of the equals sign.
x, y, and z are the unknowns; a_i, b_i, and c_i are coefficients; d_i values are constants.
After parsing, each row is converted to standard linear form so the solver can calculate determinants and classify the system.
If your coefficients are already arranged as a 3x4 augmented matrix, the adjacent 3x3 system solver is the faster workflow. This page is for equation text that still needs parsing and rearranging.
Cramer's Rule for a 3-variable system
For the regular case, the solver computes the main determinant and three replacement determinants. A nonzero main determinant means the system has exactly one ordered triple.
Delta is the main determinant, and A is the 3x3 coefficient matrix.
The coefficient matrix contains the x, y, and z coefficients from the three parsed equations.
Delta x, Delta y, and Delta z replace one coefficient column with the constants column.
Each replacement matrix swaps in the constants for the variable being solved.
When Delta is not zero, x, y, and z follow directly from Cramer's Rule.
Cramer's Rule gives a unique answer only when the main determinant is nonzero.
When the determinant is zero
If the main determinant is zero, Cramer's Rule cannot produce one unique solution. For a 3x3 system, the reliable classification comes from comparing the coefficient matrix rank with the augmented matrix rank.
A is the coefficient matrix, and A|b is the augmented matrix; equal ranks below 3 indicate dependent rows.
The system is consistent, but at least one variable is free.
A and A|b have different ranks, so the equations contradict one another.
A higher augmented rank means the system is inconsistent.
Supported input
| Supported | Not supported |
|---|---|
| + z | x² + y + |
| 2(x - 1) + 3y - | xy + |
| + y - | 1/x + y + |
| x + | sin(x) + y + |
Only linear equations in x, y, and z are supported. If one variable is missing from a row, its coefficient is treated as zero.
Geometric meaning
Each linear equation in three variables represents a plane. A unique solution is the point where three planes meet; special cases can mean no common point or infinitely many points.
Frequently Asked Questions
Sources and References
- System of EquationsWolfram MathWorld
- Cramer's RuleWolfram MathWorld
- 3x3 System of Equations SolverMath Portal
Calculations are based on the listed reference sources. Links open in a new tab.
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