x + 2 = (1/2)y = z - 3:

x + 2 = t => x = t - 2

y/2 = t => y = 2t

z - 3 = t => z = t + 3

So a vector in the direction of this line is (1, 2, 1).

So the required line will have vector equation r = (1, -1, 1) + t(1, 2, 1) and the parametric and symmetric equations follow easily.