Definition 1.1. A system of $m\phantom{\rule{mediummathspace}{0.2em}}$linear equations in variables (also called unknowns) ${x}_{1},{x}_{2},\dots ,{x}_{n}$ is any system of the following form:

$$\begin{array}{ccccccccc}{a}_{1,1}{x}_{1}& +& {a}_{1,2}{x}_{2}& +& \dots & +& {a}_{1,n}{x}_{n}& =& {b}_{1}\\ {a}_{2,1}{x}_{1}& +& {a}_{2,2}{x}_{2}& +& \dots & +& {a}_{2,n}{x}_{n}& =& {b}_{2}\\ \vdots & & & & & & & & \vdots \\ {a}_{m,1}{x}_{1}& +& {a}_{m,2}{x}_{2}& +& \dots & +& {a}_{m,n}{x}_{n}& =& {b}_{m}\end{array}$$ | (1.1) |

where ${a}_{i,j},\phantom{\rule{mediummathspace}{0.2em}}{b}_{i}$ are real numbers. We say that ${a}_{i,j}$ is the **coefficient** with the variable (unknown) ${x}_{j}$ in the $i$-th equation, and that ${b}_{i}\phantom{\rule{mediummathspace}{0.2em}}$is the **right hand side** of the $i$-th equation.

A solution to system (1.2) is a sequence ${s}_{1},{s}_{2},\dots ,{s}_{n}$ of numbers that is simultaneously a solution for each equation in the system. Solving system (1.2) means to find all the solutions.

Definition 1.2. We call the system (1.1)** consistent **if it has at least one solution, and **inconsistent **if it does not have any solution.

Example 1.3. The following system of linear equations

$$\begin{array}{ccccc}x& +& 2y& =& 7\\ x& -& y& =& 1\end{array}$$ | (1.2) |

is consistent. Indeed, $x=3$ and $y=2$ is a solution of the system (1.2), since

$$3+2.2=7,\phantom{\rule{mediummathspace}{0.2em}}3-2=1.$$ |

Example 1.4. The followig system of linear equations

$$\begin{array}{ccccc}x& +& 2y& =& 7\\ 2x& +& 4y& =& 1\end{array}$$ | (1.3) |

is inconcistent. Indeed, by the first equation we have $x+2y=7$, hence $2x+4y=14$, and by the second equation requires at the same time $2x+4y=1$.