Okay I kind of understand that, however in the example I gave, z was set to 0 in both points in both line segments because I wanted to make it as simple as possible. When I implement this solution the values for the lines could be anything.

So what you are suggesting is that I perform the gaussian matrix elimination twice, once with say x and y, and the second time with say y and z.

If both turn out to be consistent with each other, the point of intersection has been found. If they turn out to be inconsistent, there is NO point of intersection.

Is this correct?

Thankyou once again for your time.