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).