Equation says this:

$\displaystyle \frac{x-3}{-2}=\frac{y+1}{-1}=\frac{z}{3}$

But the book tells me it can also be written like this:

$\displaystyle d: \left\{\begin{matrix}

x-2y-5=0\\3y+z+3=0

\end{matrix}\right.$

I got confused on how that could be so I tried to get the second expression of a line from the first expression of the line by trying cross multyplication:

$\displaystyle \frac{x-3}{-2}=\frac{y+1}{-1} =\: \: -1(x-3)=-2(y+1)$

But I get this which is close but not it:

$\displaystyle -x+2y+5=0$

Although when I do this:

$\displaystyle \frac{y+1}{-1}=\frac{z}{3} =\: \: -z=3(y+1)$

I get something that resembles the equation in the second part:

$\displaystyle 3y+z+3=0$

Am I doing something wrong here? How does this form a line exactly because I grew up with the y = mx+b equation and this is the first time I am seeing this.