1. The problem statement, all variables and given/known data Consider the following system: x + by = -1 ax + 2y = 5 Find the conditions on a and b such that the system has no solution, one solution or infinitely many solutions. 2. Relevant equations General Algebra really. 3. The attempt at a solution Previously we had been using augmented matrices to solve these sort of problems but using one here doesn't really help. Rearranging the equations you get x = (5b + 2)/(ab - 2) and y =(5 + a) / (2 - ab) The answer is: If ab≠2 then the unique solution is x = (5b + 2)/(ab - 2) and y =(5 + a) / (2 - ab). If ab = 2 and ab ≠ -5 then there are no solutions. If ab =2 and a = -5 then x = 2t/5 -1 and y = t. I understand why ab cannot be 2 for the first part of the answer but why is the fact that a can or cannot equal 5 when ab = 2 relevant for the second and third parts of the answer? Thanks.