# 1  Systems of Linear Equations

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:

 $\begin{array}{ccccccccc}{a}_{1,1}{x}_{1}& +& {a}_{1,2}{x}_{2}& +& \dots & +& {a}_{1,n}{x}_{n}& =& {b}_{1}\\ {a}_{2,1}{x}_{1}& +& {a}_{2,2}{x}_{2}& +& \dots & +& {a}_{2,n}{x}_{n}& =& {b}_{2}\\ ⋮& & & & & & & & ⋮\\ {a}_{m,1}{x}_{1}& +& {a}_{m,2}{x}_{2}& +& \dots & +& {a}_{m,n}{x}_{n}& =& {b}_{m}\end{array}$ (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 $2x+4y=1$.