Ok so say I have equations $5x +2y = 3z = 10$ and $2x + 3y + 4z = 12$, and want to find the line of intersection.
I have solved this before by eliminating one variable to get an equation in the other two, then eliminating a different variable two get an an equation in a different two variables then I got an equation for a line through that. For the above example eliminating $x$ gives: $11y + 14z = 40$ and eliminating $y$ gives: $11x + z = 6$ and then from these equations I got $14z = 84 - 154x = 40 - 11y$ which is the equation of a line. I didn't really know if this method was correct because I didn't really understand what I was doing, I just manipulated some equations until I got an equation in the form of an equation of a line but when I tested it, I found the line is correct.
My question is when I eliminate one variable to get an equation in the other two, what does that equation actually mean? is it a plane or line or and if so, what line or what plane is it representing?
[Turning my comment into a full-fledged answer] As a general principle, whenever you have a set that is the intersection of the solution sets of two equations $f(\mathbf p)=0$ and $g(\mathbf p)=0$, then the solution set of every linear combination of the two equations also contains this set: if $f(\mathbf p)=0$ and $g(\mathbf p)=0$ for some particular $\mathbf p$, then obviously $af(\mathbf p)+bg(\mathbf p)=0$ for all $a$ and $b$.
In this particular situation, when you eliminate a variable by combining multiples of the two plane equations, you get an equation of some other plane that also passes through the intersection of the original two planes. (In fact, if the original two equations are independent, then every plane that includes their common intersection line has an equations that’s a linear combination of the first two.) So, what you’ve basically done is recast the intersection of the two planes as the intersection of two other planes, ones that have equations that are more convenient for extracting a different kind of description of the intersection line.