What does Gaussian elimination show?

It swaps rows and adds multiples of one row to another until the matrix is in row echelon form, then uses back-substitution. Every row operation and intermediate matrix is printed. This is not Gauss-Jordan reduction to reduced row echelon form.

Can I enter equations instead?

Yes. The parent tool supports equation entry as well as an augmented matrix; this page starts with the matrix form for a transparent elimination trace.

Which pivoting method is used?

Partial pivoting: the row with the largest absolute coefficient in the current column leads that column. The arithmetic uses exact fractions. Complete pivoting, scaled partial pivoting and a no-pivoting mode are not offered.

Gaussian Elimination Calculator

Enter an augmented matrix and follow each row operation to echelon form. Back-substitution then gives a unique solution or a family of solutions; a contradictory row identifies a system with no solution. The page opens with every intermediate matrix visible.

3 variables, 3 equationsAugmented matrixEvery step shown

 2x +  y -  z =    8-3x -  y + 2z =  -11-2x +  y + 2z =   -3elimination         3 row operations, 2 row swaps, pivoting on the largest coefficientback-substitution   3 stepsx = 2        y = 3        z = -1residual            0   0   0        unique solution

Enter as
Variables
Equations
Working

x: 2
y: 3
z: -1
Case: Unique

The augmented matrix

One row per equation, one column per variable, and the constant after the equals sign. A cell takes a whole number, a decimal like 0.5 or a fraction like 1/3, and a blank cell counts as 0.

xyzconstant

The working

Forward elimination first, largest coefficient leading each column, then back-substitution from the bottom row up. Every multiplier is an exact fraction.

R1 <-> R2 pivot on -3, the largest coefficient left in column x

[ -3  -1   2 | -11 ]
[  2   1  -1 |   8 ]
[ -2   1   2 |  -3 ]

R2 -> R2 + (2/3)R1 clears x from row 2

[  -3   -1    2 | -11 ]
[   0  1/3  1/3 | 2/3 ]
[  -2    1    2 |  -3 ]

R3 -> R3 - (2/3)R1 clears x from row 3

[  -3   -1    2 |  -11 ]
[   0  1/3  1/3 |  2/3 ]
[   0  5/3  2/3 | 13/3 ]

R2 <-> R3 pivot on 5/3, the largest coefficient left in column y

[  -3   -1    2 |  -11 ]
[   0  5/3  2/3 | 13/3 ]
[   0  1/3  1/3 |  2/3 ]

R3 -> R3 - (1/5)R2 clears y from row 3

[  -3   -1    2 |  -11 ]
[   0  5/3  2/3 | 13/3 ]
[   0    0  1/5 | -1/5 ]

Back-substitution

From row 3: (1/5)z = -1/5, so z = -1.
From row 2: (5/3)y + (2/3)z = 13/3 with z = -1, so y = 3.
From row 1: -3x - y + 2z = -11 with y = 3 and z = -1, so x = 2.

The check

2x + y - z = 8, residual 0
-3x - y + 2z = -11, residual 0
-2x + y + 2z = -3, residual 0

Each value is substituted back into the equations as you typed them: every residual is exactly zero, because the arithmetic is done in fractions rather than decimals.

The three answers a linear system can have

Load any of them to see which one you are looking at and why.

x + y = 3, x - y = 1two lines that cross once
x = 2, y = 1
x + y = 1, 2x + 2y = 3same line, different constant
No solution: the equations contradict each other
x + y = 3, x + y = 3one equation repeated, rank below the variable count
x = 3 - y, with y free

Three answers, and the rank tells you which. Count the rows left with a leading coefficient after elimination: that is the rank. Rank equal to the number of unknowns gives one solution. A row that reduces to 0 = something other than zero is a contradiction, so there is no solution at all. Rank below the number of unknowns leaves free variables, and every value of them is a solution, which is why the answer comes back as a formula rather than a number. Telling those apart needs zero to mean zero, so every coefficient here is held as a fraction of two whole numbers and nothing is rounded on the way.

Common questions

What does Gaussian elimination show?: It swaps rows and adds multiples of one row to another until the matrix is in row echelon form, then uses back-substitution. Every row operation and intermediate matrix is printed. This is not Gauss-Jordan reduction to reduced row echelon form.
Can I enter equations instead?: Yes. The parent tool supports equation entry as well as an augmented matrix; this page starts with the matrix form for a transparent elimination trace.
Which pivoting method is used?: Partial pivoting: the row with the largest absolute coefficient in the current column leads that column. The arithmetic uses exact fractions. Complete pivoting, scaled partial pivoting and a no-pivoting mode are not offered.

Gaussian elimination that pivots on the largest available coefficient for numerical stability, with every row operation printed so you can follow it. The solution is substituted back into the equations you typed and the residual is shown, so you can see the arithmetic close. A nonlinear term is refused by name rather than quietly linearised.