1. The problem statement, all variables and given/known data So the actual problem "Find the value of a for which the following system of linear equations has a solution" 2x + 4y + z = a -4x -7y + 0 = 1 0 -1y -2z = 1 2. Relevant equations 3. The attempt at a solution I thought one approach was to find a basis for the corresponding matrix and see what value of a would make that vector in the space formed by the basis. That is, see what value of a would make (a, 1, 1) in the range of the matrix. But when I row reduce I get 2 4 1 0 1 2 0 0 0 So I want to say the 1st and 2cnd columns form a basis -> (2, -4, 0) and (4, -7, -1). But then some linear combination of these should equal (1, 0, -2). However, the first components of the basis appear to never be able to combine linearly to 1. That is there are no integers x,y such that 2x + 4y = 1 since -> x + 2y = 1/2.....? I feel like I must be making a really trivial mistake?!?!