Let and be two lines in which are not parallel. Prove that the intersection is either empty or contains exactly one point.

*Proof.* Since and are not parallel we know . Now, suppose there exist two distinct points . This means there exist real numbers such that

Since are distinct points we also know and . Then we have

But then

for some . Thus, the lines are parallel, contradicting our assumption that the lines are not parallel. Hence, the intersection contains at most one point

Let X be the point of intersection of the two lines.

So

X=P+tA = Q+sB

Where t, s are scalars

So

P-Q=(-t) A + sB

Since A and B are not parallel, A != cB

For some scalar c, so they are linearly independent. So there exists only one way to get P-Q

So only one point.

Sorry if it is wrong.

Your solutions have helped me a lot.

Thanks