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?

In general, you can exactly determine the value of (at most) one unknown per linear equation. You have two equations, but three unknowns ($x, y, z$), and so you can (assuming they do not contradict eachother) determine two of the values in terms of the third, but the third variable will not be restricted and can take on any value. This is what we mean by infinitely many solutions.
