Confused with a system of equations with three variables that has infinitely many solutions I'm studying High School Algebra and it had this question:

Solve the system by equations: 
\begin{align*}   x + y - z &= \,0 \\    2x + 4y - 2z &= 6 \\    3x +
 6y - 3z &= \,9 \end{align*}

The solution was:

infinitely many solutions (x, 3, z) where x = z − 3; y = 3; z is any real number

I've spent hours on the problem. The textbook just gave a vague explanation and I can't seem to get how it works. Can someone please intuitively explain how this is? 

Hint: Dividing the second equation by $2$ and the third by $3$ we get
$$x+y-z=0$$
$$x+2y-z=3$$
$$x+2y-z=3$$
the second and the third equation are the same.
Multiplying the first equation by $-1$ and adding to the second we get $y=3$.
Plgging this into the first and second equation we get
$$x-z=-3$$
$$x-z=-3$$ so we obtain the solutions
$$x,3,x+3$$
