What is Normalization? I was reading Problem Solving Strategies by Arthur Engel, and there in inequality chapter, author wrote, "for homogeneous inequalities, we may take various Normalizations".
And in many problems he used it ( for example, in one he took $x+y+z =1$)  but never explained exactly what it is. I went through web but just couldn't find anything useful. ( I'm a high School student)( I do understand what homogeneous equations are.)
Can anyone please explain it simply? Thanks in advance.
 A: For homogenous equations, if $(x, y, z)$ is a nontrivial solution (assuming it's in three unknowns), then so is $(ax, ay, az)$ for all real numbers $a$. This means that we automatically know that any non-zero solution generates a whole line of solutions.
Because of this, a set of homogeneous equations will always be either underdetermined or inconsistent. If it's inconsistent, there is nothing to be done, but if it's underdetermined then our regular equation solving methods aren't really constructed to deal with that directly.
So what one usually does is to just make up some simple nonhomogenous equation and insert. Geometrically, this has the interpretation of intersecting all the lines that solve the original equation with some hypersurface so that hopefully those lines become points. This new, (hopefully) determined system of equations is something we're more equipped to solve with regular techniques.
Exactly what that new equation ought to be differs from context to context. Just something like $x = 1$ is often enough (if there happens to be a solution with $x = 0$ it will disappear, although that is also often easy to check for). I'm sure some times $x+y+z = 1$ makes complete sense as well.
