Solving a linear equation with a single unknown

Systems of linear equations and matrices: Linear equations

Solving a linear equation with a single unknown

Each linear equation can be reduced to a basic form. Given such a basic form, solving the equation is not so difficult anymore. Here we recall how this is done for a linear equation with a single unknown.

In general, the solutions of the linear equation $a\cdot x+b=0$ with unknown $x$ and real numbers $a$ and $b$ be found as follows.

$\,$ case	$\,$ solutions
$\,a\ne0\phantom{x}$	$\,$ exactly one: $x=−\dfrac{b}{a}\,$
$\,a=0$ and $b\ne0\,$	$\,$ none
$\,a=0$ and $b=0\,$	$\,$ any number $x\,$

The linear equation $3x+8=0$ has solution $x=-\frac{8}{3}$ .
The equation $0\cdot x + 8 = 0$ does not essentially contain the unknown: it is equivalent to $8=0$ , an equation which is not satisfied. There is no solution.
The equation $0\cdot x + 0 = 0$ also does not essentially contain the unknown, but it is equivalent to $0=0$ , an equation which is always satisfied. Each value of $x$ is a solution.

We explain why. (The equation is $ax+b=0$ ).

$\,$ case	$\,$ solutions	$\,$ statement
$\,a\ne0\,$	$\,$ exactly one: $x=−\dfrac{b}{a}\,$	$\,$ Subtract $b$ from both sides and next divide $\,$ them by $a$ .
$\,a=0$ and $b\ne0\,$	$\,$ none	$\,$ The equation is $b=0$ ; it is $\,$ not true, regardless of the choice of $x$ .
$\,a=0$ and $b=0\,$	$\,$ every number $x\,$	$\,$ The equation is $b=0$ ; it is true $\,$ for each choice of $x$ .

In the above rule for the solution of the linear equation $a\cdot x+b=0$ with unknown $x$ , the symbols $a$ and $b$ represent real numbers for which specific values can be entered. But in fact we have written the equation and the solution using $a$ and $b$ without writing down specific values. In other words, $a$ and $b$ are parameters.

Incidentally, all statements also apply to complex numbers instead of real.

There is no need to remember these rules, because the solutions are easy to find by reductions (it is not strictly necessary to reduce the equation to a basic form first). The three cases can also be identified geometrically in terms of lines, as we will see later. For each case we give an example.

Solve the equation $x+3=3 x+1$ in $x$ .

$x=1$

To see this, we reduce the equation as follows.

$\begin{array}{rclcl}-2 x+3&=&1&\phantom{x}&\color{blue}{\text{the term }3 x\text{ moved to the left hand side}}\\ -2 x &=&-2&\phantom{x}&\color{blue}{\text{the term }3\text{ moved to the right hand side}} \\ x &=&1&\phantom{x}&\color{blue}{\text{dividing by }-2\text{}}\tiny.\end{array}$

Hence, the only solution to the equation is $x=1$ .

New example