Expert: Richard J. Raridon - 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
Let's look at (b) first.  Yes, the lines will be parallel.  Therefore, m would equal 4 and n would be any number except for 2.  
(c) m=4, n=2
(a) m not equal to 4, n = any number
(a) easy to prove, just solve for the solution
(b) try to solve and see that you can't
(c) divide 2nd eq. by 2 to get 1st eq.