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Matrix with 2 unknowns

  1. Sep 19, 2013 #1
    1. The problem statement, all variables and given/known data
    In the following problem, find conditions on a and b such that the system has no solution, one solution, and infinitely many solutions.

    x + by = -1
    ax + 2y = 5

    2. Relevant equations
    None that I know of.

    3. The attempt at a solution
    Basically, first I put the entire equation into a matrix.
    [ 1 b | -1
    a 2 | 5 ]
    I reduce the bottom by subtracting R2 - aR1
    [ 1 b | -1
    0 2-ab| 5 +a ]
    I then reduce the bottom again by dividing R2/(2-ab)
    [ 1 b | -1
    0 1 | (5+a)/(2-ab)]
    I remove the b from the top by subtraction: R1 - bR2
    [1 0 | -1 - b((5+a)/(2-ab))
    0 1 | (5+a)/(2-ab) ]
    This leaves me with the values for x and y, and for the first question I am correct in saying if ab = 2, then there is no solution as it is undefined. However, my unique solution is somehow wrong and I would like some help in determining if I made an error or I somehow didn't reduce something.

    The correct unique solution is: x = (-2 - 5b)/(2-ab) y = (a+5)/(2-ab)

    Also, I have no idea what finding an infinite solution means, I would really like some help on clarifying that. Thank you.
  2. jcsd
  3. Sep 19, 2013 #2


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    Science Advisor
    Homework Helper

    Hi Temp0! Welcome to PF! :smile:
    That is the same as yours! :wink:
    Hint: if A and B are two solutions, what can you say about A - B ? :smile:
  4. Sep 19, 2013 #3
    For example: if you are given graphs of f(x) and g(x) [neither contain discontinuities] and g(x) 's graph never "falls under" or crosses f(x) , no matter what the argument, g(x) has always higher values than f(x) and you are asked to provide solutions for g(x) > f(x) then you can say that there are infinitely many solutions. But you are dealing with an equal sign so that must mean the 2 graphs are...?
  5. Sep 20, 2013 #4
    Hmm, thanks for the help on the infinite thing, I finally get that ^^. However, i've tried reducing and expanding my answer, but it never becomes the same as the answer in the book. Are there any hints you can give me? =D

    edit: nvm, I just looked at it again and realized how to get to the answer, thanks for your help =p.
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