Solving Equations as Proofs

On a test, a student wrote $x=9$ for this problem: $\text{Solve for x, }\sqrt{28-3x}=x-8$. I gave him a little bit of credit since that was the correct answer.  I assumed that he had guessed the answer and checked it mentally.  If he had shown me how he had checked it, I would have given him a little more credit.  Why not full credit?  Because he didn’t demonstrate in some manner that $9$ was the only answer.  The method that he was taught was to square both sides of the equation and then solve for $x$ from there. This technique for solving radical equations is taught as the method for finding solutions and that is the point of view that I took when I lectured on it, but really the technique is an argument that $9$ is the only solution.  Here is the process with the argument on the right.

Solving an Equation as a Proof

If my student had just written $x=9$ for this problem: $\text{Solve for x, }2x-3=15$, again he would have got a little bit of credit.  If he showed that the equation is true if $x=9$ he would have gotten more credit.  If he had said that linear equations have only one solution he would have gotten still more credit but not full credit since linear equations can have an infinite number of solutions, for instance, $2(x+8) = 2x+16$.  For full credit he would have to show that the equation was not an identity by possibly showing that another number does not check.

In point of fact, I generally accept the correct answer correctly checked for linear equations in the lower level algebra courses.