Could anyone help me with this proof? Thanks
Show that two distinct lines given by the equations $a_ix+b_iy+c_i=0$ for $i=1,2$ in $\mathbb R^2$ intersect if and only if $a_1b_2\ne a_2b_1$, and otherwise they are parallel in $\mathbb R^2$ but intersect at a point at infinity in $\mathbb P^2$.
The slopes of the two equations will be $-{a_1\over b_1}$ and $-{a_2\over b_2}$ respectively.
If $a_1b_2\neq a_2b_1$, then ${a_1\over b_1}\neq{a_2\over b_2}$ and $-{a_1\over b_1}\neq-{a_2\over b_2}$
Since the slopes are not equal, the lines are not parallel and must intersect.
However, if $a_1b_2 = a_2b_1$, then the slopes are equal, the lines must be parallel (since they are distinct), and they do not intersect.