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Solve The System Of Linears Equations for x and y

  1. Jun 27, 2013 #1
    1. The problem statement, all variables and given/known data
    [itex](\cos \theta )x + (\sin \theta )y = 1[/itex]

    and

    [itex](-\sin \theta )x + (\cos \theta )y = 0[/itex]

    2. Relevant equations



    3. The attempt at a solution

    Evidently the answer is that [itex]x = \cos \theta[/itex] and that [itex]y = \sin \theta[/itex].

    Here is my work:

    [itex]x = \frac{1 - (\sin \theta )y}{\cos \theta}[/itex]

    Substituting this into the second equation, and simplifying:

    [itex]y = \frac{\tan \theta}{\sin \theta tan \theta + \cos \theta}[/itex]

    I then took this equation and back-substituted into [itex]x = \frac{1 - (\sin \theta )y}{\cos \theta}[/itex], hoping that everything would simplify such that [itex]x= \cos \theta[/itex]; however, things began to look quite messy. How am I to solve this problem?
     
  2. jcsd
  3. Jun 27, 2013 #2

    LCKurtz

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    Multiply the numerator and denominator of that last equation by ##\cos\theta## and you will have it. Much easier to use determinants in the first place though.
     
  4. Jun 27, 2013 #3

    HallsofIvy

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    Personally, I would not have done the problem that way. Starting from the original equations,
    [itex]cos(\theta)x+ sin(\theta)y= 1[/itex] and
    [itex]-sin(\theta)x+ cos(\theta)y= 0[/itex]

    Multiply the first equation by [itex]cos(theta)[/itex] and the second equation by [itex]-sin(\theta)[/itex] to get
    [itex]cos^2(\theta)x+ sin(\theta)cos(\theta)y= cos(\theta)[/itex]
    [itex]sin^2(\theta)x- sin(\theta)cos(\theta)y= 0[/itex]
    and then add:
    [itex]x= cos(\theta)[/itex]

    Then multiply the first equation by [itex]sin(\theta)[/itex] and the second equation by [itex]cos(\theta)[/itex]to get [itex]sin(\theta)cos(\theta)x+ sin^2(\theta)y= sin(\theta)[/itex]
    [itex]-sin(\theta)cos(\theta)x+ cos^2(\theta)y= 0[/itex]
    Adding gives [itex]y= sin(\theta)[/itex].
     
  5. Jun 27, 2013 #4
    And so theta will be the parameter to the parametric equations that represent the solution?
     
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