Let the parameter be t.

We want a function x(t) that gives x the values -4 and -3. Well, this is supposed to make a line, so we need to make x(t) a line. I'm going to have x(t) pass through the points (t, x) = (0, -4) and (1, -3).

where . Now I will plug the point (0, -4) into this:

So my first parametric function will be .

Now we need a y(t). Again we must make y(t) linear, but now we also have to pass the function y(t) through the points (t, y) = (0, 2) and (1, 3) so that it matches what x(t) does.

So where . We easily find that , so that

.

Notice that changing the parameter t does nothing to the z coordinate of the points on the line. Thus .

So the parametric version of your line is:

(x(t), y(t), z(t)) = (t - 4, t + 2, 0).

The only difference in any other correct solution is the choice of t values I used. (Recall that I used t = 0 and t = 1.) We could easily have used and or something. That would merely change the scale of the parameter and would still produce a correct solution.

-Dan