# Any two distinct lines in the projective plane meet at a unique point

I am trying to do an exercise from Rick Miranda's book on Algebraic Curves and Riemann surfaces, where he defines

A degree one curve in the projective plane, defined by a homogeneous polynomial in x,y,z of degree one, is called a line.

Prove that any two distinct lines in the projective plane meet at one unique point and give a formula for that point.

So basically he suggests doing this by linear algebra i think, but i cant quite seem to find an invertible matrix or system that gives an invertible matrix so im kinda stuck , so any tips are aprecciated, just something to get me rolling. Thanks in advance!

A line $$L$$ in the projective plane has the equation $$ax+by+cz=0$$ where $$a,b,c$$ are not all 0. Let $$L'$$be the line $$a'x+b'y+c'z=0$$. The lines $$L$$ and $$L'$$ are distinct iff the vectors $$[a,b,c]$$ and $$[a',b',c']$$ are not proportional in which case the cross-product $$[a,b,c] \times [a',b',c']$$ represents the unique point of intersection, i.e. the point is $$(bc'-cb':-(ac'-ca'):ab'-ba')$$