1. The problem statement, all variables and given/known data In the following problem, find conditions on a and b such that the system has no solution, one solution, and infinitely many solutions. x + by = -1 ax + 2y = 5 2. Relevant equations None that I know of. 3. The attempt at a solution Basically, first I put the entire equation into a matrix. [ 1 b | -1 a 2 | 5 ] I reduce the bottom by subtracting R2 - aR1 [ 1 b | -1 0 2-ab| 5 +a ] I then reduce the bottom again by dividing R2/(2-ab) [ 1 b | -1 0 1 | (5+a)/(2-ab)] I remove the b from the top by subtraction: R1 - bR2 [1 0 | -1 - b((5+a)/(2-ab)) 0 1 | (5+a)/(2-ab) ] This leaves me with the values for x and y, and for the first question I am correct in saying if ab = 2, then there is no solution as it is undefined. However, my unique solution is somehow wrong and I would like some help in determining if I made an error or I somehow didn't reduce something. The correct unique solution is: x = (-2 - 5b)/(2-ab) y = (a+5)/(2-ab) Also, I have no idea what finding an infinite solution means, I would really like some help on clarifying that. Thank you.