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Solution Sets and Linear Equations

  1. Apr 8, 2005 #1
    Problem
    Prove that, where a, b, c, d, e are real numbers and a <> 0, if ax + by = c has the same solution set as ax + dy = e, then they are the same equation.

    Given Solution
    If a <> 0 then solution set of the first equation is {(x,y) | x = (c - by)/a}. Taking y = 0 gives the solution (c/a, 0), and since the second equation is supposed to have the same solution set, substituting into it gives a(c/a) + d(0) = e, so c = e. Then taking y = 1 in x = (c - by)/a gives a((c - b)/a) + d = e, which gives b = d. Hence they are the same equation.

    My Thoughts
    I don't buy into the solution above because it assumes that one of the members of the solution set has y = 0 and that another has y = 1. And anyways, you can take any two-variable two-equation linear system (where the equations aren't equal) and solve to get the solution set. Is this problem bogus or what?
     
  2. jcsd
  3. Apr 8, 2005 #2

    matt grime

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    Science Advisor
    Homework Helper

    No. The equations define straight lines. They are the same solution set iff they are the same line, which is iff they are "the same equation"
     
  4. Apr 8, 2005 #3
    So what you're saying is that if a != 0, then it's possible that ax + b = c (or ax = c) are not solvable (for x)?
     
  5. Apr 8, 2005 #4
    OK. I see where I was confused. This makes sense now. Cheers.
     
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