Chapter 9.3, Problem 8P

What's wrong with the next reasoning that supposedly proves that the equations $x-1=0$ has no solutions ?

Starting with the equation $x-1=0$ , divide both sides by $x-1$

   $\begin{array}{c}x-1=0\\ \left(x-1\right)/\left(x-1\right)=0/(x-1)\\ 1=0\end{array}$

Because the equation 1=0 is false, the equations $x-1=0$ is false for all values of x and therefore has no solutions.