Chapter 2 : Linear equations

1. Linear equation

A linear equation has the following form :

$$a_1x_1 + a_2x_2 + \cdots + a_nx_n = b \tag{1}$$

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

Example I

$(2)$ is a linear equation.

$$\sqrt{2}x -\frac{3}{4} y = -12 \tag{2}$$

Example II

$(3)$ is not a linear equation.

$$2x^3 + -5y = 1 \tag{3}$$

2. Linear system

A linear system (or system of linear equations) consists of several linear equations involving the same variables.

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

Example III

$$\left\{ \begin{array}{ r@{~}c@{~} r!{~} r r r r } 2x-4y & = & 2 & & & & (4) \\ x+2y & = & -7 & & & & (5) \\ \end{array} \right.$$

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 $(7)$.

$$\begin{array}{ r r r c@{~} r@{~} } (6) & & & 2x-4y & = & 2 \\ (7) & & & -2x-4y & = & 14 \\ \hline (8) & & & -8y & = & 16 \\ \end{array}$$

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

$$\left\{ \begin{array}{ r@{~} c@{~} r!{~} r r r r } -x-3y+z & = & 10 & & & & (9) \\ x+y-3z & = & -12 & & & & (10) \\ 3x+4y+2z & = & -5 & & & & (11) \\ \end{array} \right.$$

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 $(19)$.

$$\begin{array}{ r r r c@{~} r@{~} } (18) & & & -2y-2z & = & -2 \\ (19) & & & 2y+22z & = & 62 \\ \hline (20) & & & 20z & = & 60 \\ \end{array}$$

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) :

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

Wrap-up