Results 1 to 3 of 3

Math Help - Confusing system of simultaneous equations...

  1. #1
    PTL
    PTL is offline
    Junior Member
    Joined
    Aug 2009
    Posts
    29

    Confusing system of simultaneous equations...

    Find three vectors p, v, w in R^4, such that the solutions to the system
    w+3x+3y+2z=1
    2w+6x+9y+5z=5
    -w-3x+3y=5
    have the form p + \lambda v + \mu w, with \lambda, \mu in R.

    I assumed one would get rid of the z, and be left with two equations to solve three unknowns, so would write everything in terms of one vector, but simplifying the first two equations gives the third

    So now I'm completely bewildered...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    You were not taught Gauss-Jordan reduction?
    Do you have a textbook? It should have some examples of the method used to solve such systems. I find it hard to believe you were asked to find a basis for the solution space without being taught how.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Failure's Avatar
    Joined
    Jul 2009
    From
    Zürich
    Posts
    555
    Quote Originally Posted by PTL View Post
    Find three vectors p, v, w in R^4, such that the solutions to the system
    w+3x+3y+2z=1
    2w+6x+9y+5z=5
    -w-3x+3y=5
    have the form p + \lambda v + \mu w, with \lambda, \mu in R.

    I assumed one would get rid of the z, and be left with two equations to solve three unknowns, so would write everything in terms of one vector, but simplifying the first two equations gives the third

    So now I'm completely bewildered...
    Let's see. You transform the system to row-echelon form. First subtracting the first equation twice from the second and adding it to the last, like this
    \begin{array}{rcrcrcrcr|lcl}<br />
w &+& 3x &+& 3y &+& 2z &=& 1 &\cdot (-2) &|& \cdot 1\\<br />
2w &+& 6x &+& 9y &+& 5z &=& 5 &\cdot 1 & &\\<br />
-w &-& 3x &+& 3y & & &=& 5 & &| & \cdot 1\\\cline{1-9}<br />
\end{array}

    which gives (if I'm not mistaken)
    \begin{array}{rcrcrcrcr|l}<br />
w &+& 3x &+& 3y &+& 2z &=& 1 &\\<br />
&& && 3y &+& z &=& 3 &\cdot (-2)\\<br />
&& && 6y &+&2z&=& 6 &\cdot 1\\\cline{1-9}<br />
\end{array}

    Then you subtract twice the second equation from the last, which gives
    \begin{array}{rcrcrcrcr|}<br />
w &+& 3x &+& 3y &+& 2z &=& 1\\<br />
&& && 3y &+& z &=& 3\\<br />
&& && &&0&=& 0 \\\cline{1-9}<br />
\end{array}

    which is now, you guessed it, in row-echelon form.
    Now what...? - Now we do back-substitution from the last to the first equation, setting superfluous variables to arbitrary values and solving for the one remaining variable: The last equation is trivially true. In the second you may set z := \mu to an arbitrary value and solve for y, which gives y=1-\tfrac{1}{3}\mu. Finally, in the first equation, you set x :=\lambda to an arbitrary value, substitute 1-\tfrac{1}{3}\mu for y and \mu for z, respectively, and then solve for w: which gives w=-3\lambda-\mu-2.
    So we find that the general solution-vector of the given system has the form
    \begin{pmatrix}w\\x\\y\\z\end{pmatrix}=\begin{pmat  rix}-3\lambda-\mu-2\\\lambda\\1-\tfrac{1}{3}\mu\\\mu\end{pmatrix}=\begin{pmatrix}-2\\0\\1\\0\end{pmatrix}+\lambda\begin{pmatrix}-3\\1\\0\\0\end{pmatrix}+\mu\begin{pmatrix}-1\\0\\-\tfrac{1}{3}\\1\end{pmatrix}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Confusing System of Equations
    Posted in the Algebra Forum
    Replies: 4
    Last Post: May 6th 2011, 08:16 AM
  2. Replies: 1
    Last Post: March 29th 2010, 08:34 AM
  3. Solving a system of simultaneous equations
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 14th 2009, 04:55 PM
  4. Confusing System of Equations
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: March 11th 2009, 10:10 AM
  5. simultaneous system
    Posted in the Algebra Forum
    Replies: 2
    Last Post: May 7th 2008, 10:30 AM

Search Tags


/mathhelpforum @mathhelpforum