How are matrices used to represent systems of linear equations?

In numerical analysis and linear algebra, a system of linear equations is represented using matrix-vector notation as the equation $Ax=b$.1

In this representation:
* $A$ is the coefficient matrix, a rectangular array containing the coefficients of the unknowns from each equation.231
* $x$ is a column vector representing the unknown variables ($x_1,x_2,\dots ,x_n$).1
* $b$ is a column vector containing the constant terms from the right-hand side of the equations.31

For computational purposes, these systems are often represented as an augmented matrix, which incorporates the constant terms into the coefficient matrix as an additional column.4

Using matrices to represent these systems is essential for numerical analysis, as it allows for the implementation of efficient, standardized algorithms—such as Gaussian elimination—to solve systems that would be impractical to handle using manual substitution.521