How do I know that this system of equations has infinitely many solutions? \begin{cases}
2x + 4y - 2z = 0\\
\\3x + 5y      = 1
\end{cases}
My book is using this as an example of a system of equations that has infinitely many solutions, but I want to know how we can know that just from looking at the equations?

Because there are three variables and only two equations.
If you imagine $x$, $y$ and $z$ as the coordinates of a three-dimensional space, then any one equation involving one or more of $x$, $y$ and $z$ determines a plane in that space. For instance: $$0x+0y+z=2$$ determines a plane which is parallel to the $x,y$ plane but two units above it.
One equation defines a plane. Two equations define two planes: and in geometry, two planes intersect in a line. Thus these two equations define a line, and you know this just because there are two of them. 
A third equation would define another plane, and the intersection of that plane and the line you have already got would give you a point. You know this just because there are three equations.
Of course there are exceptions, which you will learn - they arise because the two planes might be parallel (in which case they do not intersect) or they might be the same (in which case their intersection is a plane, not a line). These geometrical facts correspond to simple algebraic facts about the equations.
