Show that any two distinct lines in $\Bbb P^2$ intersect in one point.
Let $L_1, L_2$ be any two distinct lines in $P^2$.
Write $L_i = V (a_iX + b_iY + c_iZ), i = 1,2$.
It suffices to show that $L_1 ∩L_2 = V(a_1X +b_1Y +c_1Z,a_2X +b_2Y +c_2Z)=V$ is a point.
Now I know that there is a projective change of coordinates $T$ such that $V^T =V(Z)$ or $=V(Y,Z)$
If $V^T =V(Y,Z)$ is an unique point.
Otherwise $V^T =V(Z)$,then can I say $L_1 =L_2$?? Why?
This is my problem. If this is true then I am done