3-Variable System of Equations Solver | 3×3
Solve systems of three linear equations with three unknowns using Gaussian elimination. Handles unique, no solution, and infinite solution cases with full step-by-step row operations.
Presets
What Is the 3-Variable System of Equations Solver | 3×3?
A 3×3 linear system represents three planes in three-dimensional space. The solution is the point (or set of points) where all three planes intersect simultaneously. Three distinct planes can intersect at exactly one point (unique solution), along a line or plane (infinitely many), or not at all (no solution — inconsistent system).
Gaussian elimination with partial pivoting is the standard algorithm: it systematically eliminates variables by forward elimination to produce an upper-triangular matrix, then recovers the solution by back-substitution. Partial pivoting swaps rows to place the largest-magnitude pivot first, which prevents division by near-zero values and improves numerical stability.
Formula
System:
a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃
Augmented matrix [A|b]:
⎡a₁ b₁ c₁ | d₁⎤
⎢a₂ b₂ c₂ | d₂⎥
⎣a₃ b₃ c₃ | d₃⎦
Gaussian elimination: Row operations until upper-triangular, then back-substitute
Solution cases: rank(A) = rank([A|b]) = 3 → unique | rank < 3 → infinite | ranks differ → none
How to Use
- 1Enter coefficients: Fill in each row: a, b, c are the x, y, z coefficients; d is the right-hand side. Negative values work.
- 2Use a preset: "Unique Solution" loads a classic example; try "No Solution" and "Infinite Solutions" to see the different output types.
- 3Solve: Click "Solve System". The result appears immediately with the x, y, z values.
- 4Check verification: Each original equation is verified — the computed x, y, z are substituted back in.
- 5View steps: Toggle "Gaussian elimination steps" to see the augmented matrix after each row operation, color-coded by variable.
Example Calculation
Classic example: 2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3
- ›Augmented matrix: ⎡2 1 −1 | 8⎤ ⎢−3 −1 2 | −11⎥ ⎣−2 1 2 | −3⎦
- ›R2 ← R2 − (−3/2)R1 → eliminate x from row 2
- ›R3 ← R3 − (−1)R1 → eliminate x from row 3
- ›Continue eliminating y from row 3
- ›Back-substitute: z = −1, then y = 3, then x = 2
- ›Solution: x = 2, y = 3, z = −1
Understanding 3-Variable System of Equations | 3×3
Applications
Three-variable systems appear in circuit analysis (Kirchhoff's laws give loop equations with unknown currents), economics (input-output models), computer graphics (finding the plane through three points), and traffic flow problems. They also arise in least-squares regression with three parameters and in finite-difference numerical methods.
Geometric Interpretation
| Rank of A | Rank of [A|b] | Solution type | Geometry |
|---|---|---|---|
| 3 | 3 | Unique solution | Three planes meet at one point |
| 2 | 2 | Infinite (line) | Three planes share a common line |
| 1 | 1 | Infinite (plane) | All three planes are the same plane |
| 2 | 3 | No solution | Three planes form a prism or parallel pair |
| 1 | 2 | No solution | Two planes are parallel |
Frequently Asked Questions
When does a 3×3 system have no solution?
A system is inconsistent when elimination produces a row like [0 0 0 | k] where k ≠ 0. This means the planes have no common point. Geometrically: two parallel planes, or three planes forming a prism.
When does a 3×3 system have infinitely many solutions?
Infinite solutions occur when at least one equation is a linear combination of others. After elimination, a row of all zeros appears. The solution is a line (one free parameter) or a plane (two), expressed in terms of a parameter t.
What is partial pivoting and why does it matter?
Partial pivoting swaps rows so the largest absolute value in each column becomes the pivot. This avoids dividing by near-zero numbers, which would amplify floating-point rounding errors. It is essential for numerical stability on ill-conditioned systems.
How is this different from the 2×2 systems solver?
The 2×2 solver uses Cramer's rule and handles two equations only. This solver uses Gaussian elimination — generalizable to n×n — and also handles degenerate cases (no solution, infinite solutions) that Cramer's rule cannot.
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