Will $2$ linear equations with $2$ unknowns always have a solution? As I am working on a problem with 3 linear equations with 2 unknowns I discover when I use any two of the equations it seems I always find a solution ok. But when I plug it into the third equation with the same two variables  , the third may or may not cause a contradiction depending if it is a solution and I am OK with that BUT I am confused on when I pick the two equations with two unknowns it seems like it has no choice but to work.  Is there something about linear algebra that makes this so and are there any conditions where it won't be the case that I will find a consistent solution using only the two equations?  My linear algebra is rusty and I am getting up to speed. These are just equations of lines and maybe the geometry would explain it but I am not sure how. Thank you. 

Each linear equation represents a line in the plane.  Most of the time two lines will intersect in one point, which is the simultaneous solution you seek.  If the two lines have exactly the same slope, they may not meet so there is no solution or they may be the same line and all the points on the line are solutions.  When you add a third equation into the mix, that is another line.  It is unlikely to go through the point that solves the first two equations, but it might.
