Expert: Steve Holleran - 5/17/2007

Question
hi i am trying this question. Find the values of m,n E R for which the system of equations x+2y=1; 2x+my=n  (a)possesses a unique solution (b)is inconsistent (c) possesses infinitely many solutions. I know that for uniqueness these lines must intersect; for (b) the lines will not intersect(will be parallel i think) and for (c) the two lines will have similar equations (or will basically be the same line) HOWEVER how do i go about proving the above conditions given the equations? thank you so very much in advance, i am greatly appreciative.  

Answer
Hi Jon,

You certainly have the understanding of the problem correct for each of the cases.  As far as Proving the cases, the only thing I can come up with is this:

For the infinitely many solutions case,

if              x + 2y = 1

              2x + my = n     is the system, and if we multiply the top one by 2 and the bottom one by -1, then

              2x + 4y = 2

             -2x - my = -n

and we want both sides to be zero, so this will occur if


             m = 4 and n = 2.

For no solutions, you would want m = 4, but n = anything other than 2.

For a unique solution, I would think that you would want m not = 4, and then n could be any value.

That seems to be the best I can come up with... hope some of it helps out, even  a little

Steve Holleran