How to know which variables to parametize in a large matrix?

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(dont want anyone to solve the problem, just don't understand one thing)

So I have a homework problem where I got a 3x6 matrix, and I have to parametrize the equations and solve for each variable in terms of s t and u. Theres a decent amount of zero spaces in the matrix, and it was simple to figure out what 3 variables I could choose to solve the problem.

However when I did it, I got the wrong answer. Then it occurred to me that I could have used a different 3 variables to parametrize the matrix. And doing so would get me a different answer, naturally.

Since I have to submit my homework into an online module that just compares what I input with the key, I guess there must be a way to tell WHICH variable set, specifically, I'm supposed to use.

Any ideas? Thanks, I've looked at it and I don't see how to tell.

https://i.stack.imgur.com/8I0KW.png

(I don't want anyone to solve the problem only posting because i want to give an example... I could solve by setting the X subset 1,4,5; or 1,4,6; etc as my s t and u. Doesn't seem to make a difference I can solve the problem no matter what but get different answers)