UTM Ideals Varieties and Algorithm - Chapter 1 Section 3 Exercise 13

Problem:

Solution:

The direction of the normal is $ (1, 1, -1) \times (-1, 2, 1) $. The cross product is (3, 0, 3).
Therefore the solution is $ 3x + 3z = 0 $, or simply $ x + z = 0 $.