What is wrong with substituting a constant in a system of equations?

Question

Imagine that I have the following system of equations:

\begin{equation} \left\{ \begin{aligned} x + 2 &= 3 \\ y - 5 &= 3 \end{aligned} \right. \end{equation}

The solution (x=1, y=8) of this equation is trivial (equations do not even depend on each other).

But imagine that to solve this system I do the following:

1. I note that both: x+2 and y-5 equal 3

2. I do the substitution: x+2 = y-5

3. I arrive at the solution: y-x=7

This solution includes the correct solution of (x=1, y=8), but generally is not correct.

So my question is: Is it correct to do a substitution like this? If not, what exactly makes it wrong?

Thank you!

Answer 1

Such a substitution is logically correct. However, it is a conclusion and the result is a necessary condition that is no longer sufficient. $$ (x,y)=(1,8)\Longleftrightarrow x+2=3 \text{ AND }y-5=3\;\Longrightarrow y-x=7 \not\Longrightarrow (x,y)=(1,8) $$ Hence, the substitution gives more solutions than the two equations alone. $y-x=7$ is necessary for any solution of the two equations. However, it is no longer equivalent to both equations, i.e. we lost information.