elementary number theory - Solve $ax equiv b mod m$ without Linear Congruence Theorem or Euclid's Algorithm?

Origin page 5. The overhead doesn't look like Linear Congruence Theorem or anything from Euclid's Algorithm. page 4 tries to delineate
Casting Out the Modulus? Is this it?

So if $d|m,$ then I rewrite $\begin{align} ax & \equiv b \mod m

\\dx + x & \equiv
\\ \implies x & \equiv \end{align}$

Is this errorless? Did I blight anything?