A linear system can be written compactly as a single matrix equation

$$A\mathbf x=\mathbf b.$$

Here $A$ contains the coefficients, $\mathbf x$ contains the unknowns and $\mathbf b$ contains the constant terms.

From equations to matrices

The system

$$\begin{aligned} 2x+y&=5,\ -x+3y&=4 \end{aligned}$$

becomes

$$\begin{pmatrix} 2&1\ -1&3 \end{pmatrix} \begin{pmatrix}x\y\end{pmatrix}

\begin{pmatrix}5\4\end{pmatrix}.$$

Multiplying the matrix by the vector reproduces the two original equations.

Coefficient and augmented matrices

The coefficient matrix is

$$A=\begin{pmatrix}2&1\-1&3\end{pmatrix}.$$

For solving the system, the coefficients and constants can be placed together in the augmented matrix

$$\left[\begin{array}{cc|c} 2&1&5\ -1&3&4 \end{array}\right].$$

The vertical bar is only a visual separator; it keeps the constant column associated with the equations while row operations are performed.

Matrix notation removes repeated variable names and exposes the structure that Gaussian elimination will manipulate.