The following issue came up in my Intro to Proofs course and I wasn't sure how to explain my distaste of the student proof.
Prove that the solution for $x$ in $ax+b=c$ is unique ($a \neq 0$).
Student Proof: Solving gives \begin{align*} ax+b &= c\\ ax &=c-b\\ x &= \frac{c-b}{a}. \end{align*} Hence the solution must be $x=\frac{c-b}{a}.$
The Proof I Wanted: Suppose both $x$ and $y$ solve the equation, then \begin{align*} ax+b &= ay+b\\ ax&=ay\\ x&=y. \end{align*} Thus the solution is unique.
Is the student solution acceptable?
I feel like finding an algebraic solution is not a proof of uniqueness. My gut says that there should be a good example where algebraically solving implies a unique solution when really there isn't one. (But I can only think of simple cases where you forget a $\pm$ on a square root.)
Another way of explaining my distaste for this is that when solving choices are made to arrive at $x$. Maybe other choices would've gotten you to a different solution. Whereas in the desired proof since we assumed two solutions from the start, it's okay to make algebraic choices to show the two solutions are the same. But maybe I'm being too pedantic.
Am I justified in disliking the student solution?