Recently while doing some math problems, me and my friend came up with some equations. I forgot what the real problem was, but lets say for a moment the equation was more like this: $\omega = \psi + 1$
Let just say we need to solve this equation and my friend was like there is term "1" in the equation and we could simple subsitute $1 = \frac{\psi}{\psi}$ so the equation would turn out to be something like :
$$\omega = \psi + \frac{\psi}{\psi}.$$
So my argument was we are not allowed to put additional variable into the equation like he did there because that would bring more solutions in the end when we solve for the equation.
So I just wanted to know if we could do something like that in any equations? (Especially substituting 1 in terms of fractions with a variable and then solving for the solution)
I hope I cleared my confusion. Thanks for the help.
Assuming we're working with numbers here, there's no reason why this would not be valid, provided you never subsitute $0$ in for the new variable. The new equation is still true and should not have any more or fewer solutions.