The system with the augmented matrix 2 3 2 4 6 a ! can u explain how u figure this out? (a) has exactly one solution for any a; (b) has exactly one solution if a 6= 4; (c) has no solution if a = 4; (d) has innitely many solutions if a = 4; (e) none of the above Solution I\'m presuming your matrix looks like this: [2 3 | 2] [4 6 | a] If so, then let\'s look at our possibilities. If not, please let me know in a comment and I\'ll fix it accordingly. Every augmented matrix has a corresponding system of linear equations. This corresponds to the following system of linear equations: 2x+3y=2 4x+6y=a. Now, if we multiply the first equation by 2 on both sides, we have: 4x+6y=4 4x+6y=a. It can be verified that these are linear equations. I suggest that you graph these for different a. If a=4, then the above system consists of two instances of the same equation, which, when graphed, gives the same line: 4x+6y=4 4x+6y=4. Thus, there are infinitely many solutions to that. For example, (0, 2/3), (1, 0), and (-1, 4/3) are solutions. If a is not 4, then the above system consists of two different linear equations, which, when graphed, gives two parallel lines. For example: 4x+6y=4 4x+6y=6 Thus, these have no solution. Given the choices, the answer is d). .