An apparent paradox: a different solution when transforming a linear equation

Question

If we have $y=a(y-x)+b$, and we set $a=1$, the solution of that equation will be $x=b$.

However, if we rewrite the equation as $y=\frac{a}{a-1}x-\frac{b}{a-1}$, and again we set $a=1$, we would get that both minuend and subtrahend would tend to infinity, since $\frac{a}{a-1} \rightarrow \infty$ and $\frac{b}{a-1} \rightarrow \infty$. We would therefore get something like $(\infty \cdot x - \infty)$.

Could you explain me how to solve or clarify this apparent paradox?

Answer 1

When you rewrite the equation, you divide both sides by $a - 1$. This is not defined when $a = 1$, so you can only use the transformed equation when $a \neq 1$.

Answer 2

Transforming a linear equation using division often creates undefined or missing solutions for specific values of x. For example, a=b$\implies$$\frac{a}{b}$=1, where if b=0, a=0 if a=b, but if we rephrase it to $\frac{a}{b}$=1, $\frac{a}{b}$ is undefined.

Answer 3

Do not divide by zero. In particular, do not set $a=1$ in $\frac{a}{a-1}$.