Results 1 to 2 of 2

Thread: System of linear algebraic equations

  1. #1
    Newbie
    Joined
    Oct 2013
    From
    Croatia
    Posts
    15

    System of linear algebraic equations

    Hi, i've been trying to solve following system for last two hours and cant find a solution. Can anyone help?

    Problem: Detremnie the values of k, for which the following system has a nontrivial solution:

    9x-3y=kx
    -3x+12y-3z=ky
    -3y+9z=kz


    Find a nontrivial solution in each case.

    Thanks in advance
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2010
    Posts
    3,455
    Thanks
    1368

    Re: System of linear algebraic equations

    Solve the first and last equation for y.

    From the first equation: $\displaystyle y = \dfrac{9-k}{3}x$

    From the last equation: $\displaystyle y = \dfrac{9-k}{3}z$

    Either $\displaystyle y=0,k=9$ or $\displaystyle x=z$. If $\displaystyle y=0,k=9$ then $\displaystyle -3x-3z=0$ implies $\displaystyle x=-z$, so the solution is nontrivial (although there are an infinite number of solutions where $\displaystyle k=9$).

    Next, suppose $\displaystyle k\neq 9$ and $\displaystyle x=z$. Then plug in what you know $\displaystyle y=\dfrac{9-k}{3}x, z=x$ into the second equation:
    $\displaystyle -3x + (12-k)\dfrac{9-k}{3}x-3x = 0$

    $\displaystyle (12-k)\dfrac{9-k}{3}x = 6x$

    If $\displaystyle x=z=0$, then $\displaystyle y=0$, and this solution is trivial. So, suppose $\displaystyle x\neq 0$. Then multiply both sides by $\displaystyle \dfrac{3}{x}$. Hence $\displaystyle (12-k)(9-k) = 18$. Solve this quadratic: $\displaystyle 108 - 21k + k^2 = 18$. So $\displaystyle k^2-21k + 90 = (k-6)(k-15)=0$. So, you get $\displaystyle k=6$ or $\displaystyle k=15$.

    Hence, the three values for $\displaystyle k$ that give nontrivial solutions for the system are $\displaystyle k=6, k=9, k=15$. Note that for any of those three values for $\displaystyle k$, there are an infinite number of solutions for $\displaystyle x,y,z$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: Mar 6th 2013, 04:42 AM
  2. Replies: 2
    Last Post: Mar 21st 2011, 04:17 PM
  3. Replies: 2
    Last Post: Apr 20th 2010, 03:26 PM
  4. a nonlinear system of algebraic equations
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Mar 18th 2009, 02:44 PM
  5. System of Linear Equations
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Mar 29th 2008, 05:41 PM

Search Tags


/mathhelpforum @mathhelpforum