We use parameters if there are going to be infinite solutions. When are there infinite solutions?
* if you've got more variables to solve for, than you have equations to solve with (and thus, you will have to use replace one of the variables with a parameter)
You can actually have 2 equation in 3 unknowns and no solutions (parallel planes).
Or you can have more equations than variables and still have infinite solutions. Think of 3 or more planes all intersecting along a straight line (in a star formation). That system also has infinite solutions.
However, in Methods 3/4 you will only be asked to deal with the 2 most common situations.
1) 3 planes intersecting at a point (unique solution)
2) 2 planes intersecting at a line (infinite solutions, requiring a single parameter)
EDIT: just had a look at the Essentials textbook, and it turns out that Worked Example 12, in section 2.7 is in fact ....
3) 3 planes intersecting at a line (infinite solutions, requiring a single parameter). This is the star formation scenario.