# System of linear equations solver

> Solve 2 to 5 simultaneous linear equations exactly by Gauss–Jordan elimination, with every row operation shown, and detect no-solution or infinite cases.

Interactive version: https://www.calcopenly.com/math/system-of-linear-equations-solver
Subject: Math calculators

A system of linear equations is written as an augmented matrix [A | b], one row per equation. Gauss–Jordan elimination scales each pivot row so the pivot is 1 and subtracts multiples of it from every other row until the coefficients form the identity matrix, leaving the solution in the last column. The arithmetic uses exact fractions, so 2x + 3y = 7, 4x − y = 1 gives x = 5/7 and y = 13/7 rather than rounded decimals.

Simultaneous equations come up in circuit analysis, mixture and pricing problems, and algebra courses. The default is the 3 × 3 example from Wikipedia's Gaussian elimination article, 2x + y − z = 8, −3x − y + 2z = −11, −2x + y + 2z = −3, with the unique solution x = 2, y = 3, z = −1.

By the Rouché–Capelli theorem the system has one solution when the rank of A equals the number of unknowns, infinitely many when A and [A | b] share a smaller rank, and none when their ranks differ.

## Inputs

- **Equations as rows of numbers**: One equation per line: the coefficients, then the constant. 2 1 -1 8 means 2x + y − z = 8. A | or = before the constant is fine; fractions like 1/2 work.

## Results

- Solution — main result
- Type of system
- x₁ (x)
- x₂ (y)
- x₃ (z)
- x₄
- x₅
- Determinant of the coefficients
- Condition number (∞-norm)

## Formula

$$
\begin{gathered} [A \mid \mathbf b] \xrightarrow{\text{row operations}} [I \mid \mathbf x] \\[10pt] \kappa_\infty(A) = \|A\|_\infty \, \|A^{-1}\|_\infty \end{gathered}
$$

## Worked examples

### Wikipedia 3×3 example

- Equations as rows of numbers: 2 1 -1 8 / -3 -1 2 -11 / -2 1 2 -3
- **Solution: x = 2, y = 3, z = −1**
- **x₁ (x): 2**
- **x₂ (y): 3**
- **x₃ (z): -1**
- **Determinant of the coefficients: -1**
- **Condition number (∞-norm): 60**
- Checked against: Wikipedia “Gaussian elimination” example; Python fractions: det = −1, ‖A‖∞ = 6, ‖A⁻¹‖∞ = 10

### Fractional answer: 2x + 3y = 7, 4x − y = 1

- Equations as rows of numbers: 2 3 7 / 4 -1 1
- **Solution: x = 5/7, y = 13/7**
- **x₁ (x): 0.7142857143**
- **Determinant of the coefficients: -14**
- Checked against: Python fractions: x = 10/14, y = 4x − 1 = 13/7

### Inconsistent: x + y = 1, 2x + 2y = 3

- Equations as rows of numbers: 1 1 1 / 2 2 3
- **Type of system: No solution — the equations are inconsistent**
- **Solution: No solution**
- Checked against: 2 × (first) gives 2x + 2y = 2 ≠ 3

### Dependent equations (edge case)

- Equations as rows of numbers: 1 1 1 6 / 2 2 2 12 / 1 -1 0 0
- **Type of system: Infinitely many solutions**
- **Solution: x = 3 − t/2, y = 3 − t/2, z = t**
- **Determinant of the coefficients: 0**
- Checked against: Row 2 = 2 × row 1; x = y and 2y + z = 6 (hand elimination)

### 4×4 system

- Equations as rows of numbers: 2 1 -1 3 9 / 1 -3 2 1 17 / 3 2 1 -1 -2 / 1 1 1 1 6
- **Solution: x₁ = 1, x₂ = −2, x₃ = 3, x₄ = 4**
- **x₄: 4**
- Checked against: Constants built from x = (1, −2, 3, 4); Python fractions Gauss–Jordan confirms the unique solution

## Questions

### How do you solve a system of three equations with three unknowns?

Eliminate one variable at a time. Gauss–Jordan elimination writes the equations as a matrix, turns the first coefficient into 1, subtracts multiples of that row to clear x from the other equations, then repeats for y and z. For the default system three rounds of row operations leave x = 2, y = 3, z = −1; substituting into 2x + y − z = 8 gives 4 + 3 + 1 = 8.

### How can you tell if a system has no solution or infinitely many?

Row-reduce it and look at rows whose coefficients all become 0. A row reading 0 = 1, or 0 equal to any non-zero number, means no solution: x + y = 1 and 2x + 2y = 3 are parallel lines. A row reading 0 = 0 means one equation repeats another, leaving a free variable and infinitely many solutions. With as many equations as unknowns, a determinant of 0 signals one of these two cases.

### What is Cramer's rule?

Cramer's rule gives each unknown as a ratio of determinants: x = det(Aₓ)/det(A), where Aₓ is A with its x column replaced by the constants. For 2x + 3y = 7, 4x − y = 1, det(A) = 2 × (−1) − 3 × 4 = −14 and det(Aₓ) = 7 × (−1) − 3 × 1 = −10, so x = 10/14 = 5/7. It needs det(A) ≠ 0 and much more arithmetic than elimination beyond 3 × 3.

### What is the difference between Gaussian and Gauss–Jordan elimination?

Gaussian elimination stops at row echelon form, a triangular matrix, and finds the unknowns by back-substitution from the last equation upward. Gauss–Jordan continues until the coefficients form the identity matrix, so each row reads off one unknown directly. Gauss–Jordan takes about n³ arithmetic operations for n equations against about 2n³/3, in exchange for skipping the substitution step.

### What does the condition number of a system mean?

It bounds how much a small relative change in the coefficients or constants can grow in the solution. The ∞-norm version is κ = ‖A‖∞ × ‖A⁻¹‖∞; for the default system ‖A‖∞ = 6 and ‖A⁻¹‖∞ = 10, so κ = 60. As a rule of thumb about log₁₀ κ significant digits are lost, here about 2, and a value near 10¹² means the equations are nearly dependent.

### How accurate is the system of linear equations solver?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 5 worked examples whose answers come from independent sources; for example, “Wikipedia 3×3 example” is checked against Wikipedia “Gaussian elimination” example; Python fractions: det = −1, ‖A‖∞ = 6, ‖A⁻¹‖∞ = 10.

### Where does the method come from?

Wikipedia — Gaussian elimination (worked 3×3 example); Wolfram MathWorld — Rouché–Capelli (Kronecker–Capelli) theorem: consistency by rank.

## Sources

- [Wikipedia — Gaussian elimination (worked 3×3 example)](https://en.wikipedia.org/wiki/Gaussian_elimination)
- [Wolfram MathWorld — Rouché–Capelli (Kronecker–Capelli) theorem: consistency by rank](https://mathworld.wolfram.com/Rouche-CapelliTheorem.html)