I'd just solve

x + 2z = 0 .... (1)

2x - 3y = 4 .... (2)

simultaneously:

Let x = t. Then y = (2t - 4)/3 and z = -t/2. This is the parametric equation of a line.

Your solution isnotconsistent with mine eg. My line does not pass through (0, 0, -4/3) but yours does (in fact, that point does not even lie in the second plane ....)