Linear equation/One variable/Solution behavior/Remark

The "mother of all systems of linear equations“ is just one linear equation in one variable of the form

${}ax=b\,,$

with given elements ${}a,b$ from a field ${}K$ and wanted ${}x$. We have three possibilities how the solution behavior might look like. For ${}a\neq 0$, we can multiply the equation with the inverse ${}a^{-1}$ of ${}a$, yielding the unique solution

${}x=ba^{-1}={\frac {b}{a}}\,.$

Computationally, one can find the solution, as long as one can find the inverse element and can perform the multiplication in the field. For ${}a=0$, the solution behavior depends on ${}b$. If ${}b=0$, then every ${}x\in K$ is a solution; if ${}b\neq 0$, then there is no solution.