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Constraint on a linear system?

  1. Nov 6, 2012 #1
    The question goes like this.
    Among all solutions that satisfy the homogeneous system
    x + 2y + z = 0
    2x + 4y + z = 0
    x + 2y − z = 0
    determine those that also satisfy the nonlinear constraint y − xy = 2z

    I know that one of the solutions [0,0,0] but i'm not sure how to find the others. I row reduced the linear system and its general solution is x=y[-2,1,0].
     
  2. jcsd
  3. Nov 6, 2012 #2

    Mark44

    Staff: Mentor

    So the general solution to your system of equations is t<-2, 1, 0>, where t is any real number. Another way to write this is <-2t, t, 0>. Does this satisfy your constraint for some value of t?
     
  4. Nov 6, 2012 #3
    oh -1/2<-2,1,0>=<1,-1/2,0> is another solution to the constraint.
     
  5. Nov 6, 2012 #4

    Mark44

    Staff: Mentor

    Are you sure?
     
  6. Nov 6, 2012 #5
    i think so.
     
  7. Nov 7, 2012 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    You have already said that any solution to the linear equations must have x= -2t, y= t, z= 0. (I did not check that.)

    Now you also want to require that y- xy= 2z which is just t- (2t)(t)= 0 or [itex]t- 2t^2= 0[/tex]. That's a quadratic equation and has two solutions.
     
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