If I make a substitution e.g. "let $x = 2y$", do I need to express all "$x$" terms as "$y$" or can I have both at the same time?
For example with $x^2+x+2=5$, can I express it as $x^2+2y+2=5$, or does it have to be $(2y)^2+2y+2=5$
Note: I'm not asking this for the purpose of solving this problem, but to understand mathematical form.
On
The whole point of a substitution is to simplify the problem, not to add complexity.
This is an example of how a substitution may be used. Consider the problem $$\text{ factor } 9x^2+6x-7$$ One way of eliminating the $9$, is just to make the substitution $y=3x$, giving us $$(3x)^2+2(3x)-7 = y^2+2y-7$$ continuing we have $$y^2+2y-7 = y^2+2y+1-1-7 = (y+1)^2 - 8 = (y+1+\sqrt{8})(y+1-\sqrt{8})$$ Finally, back-substituting, we have $$(3x+1+\sqrt{8})(3x+1-\sqrt{8})$$
You don't have to, but you almost certainly want to. $x^2+2y+2=5$ is a valid substitution, but you now have two variables where you had one. This is unlikely to be progress.