easy way of finding line of intersection of 2 planes

I may be confused so I am hoping someone could verify what I am saying is correct.

To find the line of intersection of 2 planes subtract one from the other so that one of the variables cancels out. Then introduce a parameter and then solve for x y and z.

e.g.

$\displaystyle x+2y-z=10$ and $\displaystyle 2x+3y+z=0$. Subtracting 2 times the first equation from the second gives $\displaystyle -y+3z=-20$. Let z = t. Solving the equation for y give y=20+3t. Then from the first equation x=10-2(20+3t)+t

am I correct in believing that I have found the parametric equation of the line of intersection of the 2 plains mentioned above?

The reason I'm doubtful is because when I google searched "line of intersection of two planes" I found a more difficult approach that $\displaystyle r=r_o+tv$ where the cross product is used to find v. I guess this way is used to find the vector equation of the line?