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? 

If $x,y,x\in\mathbb R$ are such that $2x+4y-2z=6$, then $\frac32(2x+4y-2z)=\frac32\times6$; in other words, $3x+6y-3z=9$, which is the thir equation. Therefore, solving that system is the same thing as solving the system$$\left\{\begin{array}{l}x+y-z=0\\2x+4y-2z=6,\end{array}\right.$$which is equivalent to$$\left\{\begin{array}{l}x+y=z\\2x+4y=2z+6.\end{array}\right.$$Solving this system, you will get the solutions that you mentioned.
