Simple fractions

$\begin{aligned} \dfrac{a}{b} &= \dfrac{a}{c} \\ ac &= ab \\ ac -ab &= 0 \\ a(c -b) &= 0 \\ \\ c = b \quad &\text{OR} \quad a = 0 \end{aligned}$

Therefore it is not necessarily true every time for $c = b$

What we usually do:

$\frac{a}{b}=\frac{a}{c}$ Divide both sides by $a$ ......... $[1]$ $\frac{1}{b}=\frac{1}{c}$ Take reciprocals on both sides........ $[2]$ $b=c$

What we does is that we assume that $a\cancel{=}0$ in $[1]$ .

But when we consider all real numbers we also include $a=0$ , for which it is not true.