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# Prove that the intersection of two non-parallel lines is either empty or contains exactly one point

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 ### One comment

1. Shibasis Patnaik says:

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