How do I work with the parametric equation of a line?

Answer

Very simple: you can eliminate that parameter k three times:

k = x – 3

k = 1/2 ( y 3 )

k= z – 1

With this you can even make three planes that contain the straight line, by equating two of the right-hand sides each time. You can do that, because they are all equal to k:

x – 3 = 1/2 ( y – 3)

x – 3 = z – 1

z – 1 = 1/2 (y – 3)

These are 3 planes containing the straight line. Choose only two, write them after an accolade and your straight line is written as the intersection of two planes.

For example, do you have the parameter form (where a k is “missing” by chance):
x = 3 + k
y = 2
z = 2 – k

then you just eliminate k from x and z and you take y = 2 as the 2nd plane containing the line (which it actually does effectively. This eventually gives as line

x + z – 5 = 0
y = 2

Note that the equation of a straight line as the intersection of two boxes is therefore not unique: after all, there are infinitely many planes passing through the line, and you can choose two of them. It doesn’t matter (except maybe for the math…) which two, as long as they’re different.
The parametric equation is also not unique, because it is based on a point on the line (there are an infinite number of such points) and the direction numbers of the line (you can always divide or multiply these by an equal number, that remains the same direction) . Always keep that in mind when you solve an exercise with the solution listed. So your solution may look different, but still be just as correct.