Finding all rows of 2 variables using Gaussian Elimination


Suppose I have a system of linear equations. Using Gaussian elimination, I can determine whether a solution exists, and even find a valid solution.

During the elimination, I can combine rows together, to produce new rows with different number of variables. Is there a method to find all possible rows that contain exactly 2 variables? For example, I may want to find all equalities between variables. This is equivalent to finding all rows that contain exactly 2 variables. Is it possible to do this without trying all (exponentially many) combinations of rows?

For example if I have:

Row 1: A xor B xor C = 1

Row 2: A xor B xor D = 1

I can combine row 1 and row 2 to say that C xor D = 0

If I have a large amount of rows, and they require large combinations of large rows to produce smaller rows, is it trivial or hard to find all rows of size 2? Can I do better than adding random pairs to the system and checking it still has a solution?