By means of a finite sequence of elementary row operations, called Gaussian elimination, any matrix can be transformed to a row echelon form.  This transformation is necessary for solving a system of linear equations.  

Before getting into more detail, there are a couple of key terms that should be mentioned:

• Augmented matrix: an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices.
• Upper triangle form: A square matrix is called upper triangular if all the entries below the main diagonal are zero. A triangular matrix is one that is either lower triangular or upper triangular. A matrix that is both upper and lower triangular is a diagonal matrix.
• Elementary row operations: Swap rows, add rows or multiply rows.

Gaussian Elimination 

1. Write the augmented matrix for the linear equations.
2. Use elementary row operations on the augmented matrix $[A|b]$ to transform $A$ to upper triangle form. If a zero is on the diagonal, switch the rows until a nonzero is in its place.
3. Use back substitution to find the solution.

Example 1:  Solve the system by Gaussian Elimination:  

$\displaystyle 2x+y-z=8\\ -3x-y+2z=-11\\ -2x+y+2z=-3$

Write the augmented matrix: 

$\left[\begin{array}{rrr|r} 2 & 1 & -1 & 8 \\ -3 & -1 & 2 & -11 \\ -2 & 1 & 2 & -3 \end{array} \right] $

Use elementary row operations to reduce the matrix to reduced row echelon form:

$\left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -1 \end{array} \right] $

Using elementary row operations to obtain reduced row echelon form ('rref' in the calculator) the solution to the system is revealed in the last column: $x=2, y=3, z=-1$.