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When is an equation linear?

  1. Oct 27, 2013 #1
    1. The problem statement, all variables and given/known data

    Let k [itex]\in[/itex] [itex]\Re[/itex] / {0}. Which of the three equations is linear in x, y, z?

    x + y - z = tan(k)

    kx - (1/k)y = 6

    (3^k)x + y + z = 12


    Well the question is basic, but I'm a little stumped because the questions asks for a single linear equation. By definition a linear equation is linear if it contains only terms that are constants or products of constants and variables in the first power. By that definition I think all should be linear in x, y and z. E.g. since k is a constant, tan(k) would also be a constant, right? The second equation is also linear, but is it linear in all three variables? I think it is linear in all three since it basically also includes the product of z and the constant 0.
    And I think the third equation is similar as well, if k is supposed to be a constant.

    Can anybody just quickly tell me if there really is only one linear equation here and whether what I have come up with is correct?
     
  2. jcsd
  3. Oct 27, 2013 #2

    Mark44

    Staff: Mentor

    All three are linear. All three represent planes in R3. The fact that the equations include tan(k), 1/k, and 3k respectively, has no bearing. As far as x, y, and z are concerned, k is a constant, so tan(k), 1/k, and 3k are constants as well.
     
  4. Oct 27, 2013 #3
    Thanks a lot for the quick answer. One more question though, what does it mean for an equation to be linear in x, y and z?
     
  5. Oct 27, 2013 #4

    Ray Vickson

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    Science Advisor
    Homework Helper

    You already explained that yourself in your previous response.
     
  6. Oct 27, 2013 #5

    Mentallic

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    What if [itex]k=\pi/2+n\pi, n\in\mathbb{Z}[/itex] ? We have a problem with [itex]\tan{k}[/itex] in that situation.
     
  7. Oct 28, 2013 #6

    Mark44

    Staff: Mentor

    Sure, tan(k) is undefined for these values, but I don't think that affects the "linearness" of the equation, with regard to x, y, and z.
     
  8. Oct 28, 2013 #7

    Mentallic

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    Sure, I guess, but notice that they defined [itex]k\in\mathbb{R}/\{0\}[/itex]
    I'd hazard a guess that they excluded k=0 because of this equation

    But is it because of the kx term or the undefined y/k term? Or possibly both? If it's because of the undefined value, then I'd have to say that they probably just forgot to remove the values of k where tan(k) is undefined.
     
  9. Oct 28, 2013 #8

    Mark44

    Staff: Mentor

    That would be my guess as well. The problem writer might have gotten sloppy.
     
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