## 1. Linear equation

A linear equation has the following form :

In $(1)$, $a_i$ are called the **coefficients** and $x_i$ are called the **variables** (also called unknowns).

#### Example I

$(2)$ is a linear equation.

#### Example II

$(3)$ is not a linear equation.

## 2. Linear system

A **linear system** (also called system of linear equations) gathers several linear equations which are related to each other.

There are many methods for solving such systems. We are going to see the **addition** method only whose goal is to **eliminate** a variable by adding two equations.

#### Example III

In order to eliminate a variable by adding $(4)$ and $(5)$ together, we can for instance multiply them by $1$ and $-2$, respectively. By doing so, we get $(6)$ and $(7)$.

Let’s solve $(8)$ :

$ -8y = 16 \iff$ $y = -2$

Let’s replace $y$, for example in $(5)$ (it could be any other) :

$ x +2\cdot \underbrace{-2}_{y} = -7 \iff x -4 = -7 \iff $ $x = -3$

#### Example IV

Let’s eliminate $x$ by multiplying $(9)$ by $1$ and $(10)$ by $1$, and thus getting $(12)$ and $(13)$ (below left). Let’s one more time eliminate $x$ by multiplying $(10)$ by $-3$ and $(11)$ by $1$, and thus getting ($15$) and ($16$) (below right).

Let’s eliminate $y$ by multiplying $(14)$ by $1$ and $(17)$ by $2$. By doing so, we get $(18)$ and $(19)$.

Let’s solve $(20)$ :

$20z = 60 \iff$ $z=3$

Let’s replace $z$ for example in $(17)$ :

$ y + 11\cdot \underbrace{3}_{z} = 31 \iff y +33 = 31 \iff$ $y = -2$

Let’s replace $y$ and $z$ for example in $(10)$ (it could be any other) :

$x + \underbrace{-2}_{y} -3\cdot \underbrace{3}_{z} = -12 \iff x + -11 = -12 \iff$ $x = -1$

## Recapitulation

The point is to eliminate a variable by **adding** two equations together. For that purpose it is necessary to find the right **factors**. The addition generates a new equation which in turns can be used for the elimination of another variable.