My Math Forum Homogenous system with more variables than equations

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 October 20th, 2013, 03:47 AM #1 Newbie   Joined: Dec 2012 Posts: 22 Thanks: 0 Homogenous system with more variables than equations Did I solve this correctly? Solve the system. 3u - v + w -5x - y = 0 6u - 2v + 2w - 9x + y = 0 -9u + 3v - 3w + 11x -y = 0 Augmented matrix: $\left[ \begin{array}{cccc} 3=&-1=&1=&-5=&-1=&0 \\ 6=&-2=&2=&-9=&1=&0 \\ -9=&3=&-3=&11=&-1=&0 \end{array} \right]= \left[ \begin{array}{cccc} 1 &\frac{-1}{3}=&\frac{1}{3}=$ My answer: u - 1/3v + 1/3 w = 0 u = 1/3(v - w) where v and w are any real numbers. v = 3u + w, where u and w are any real numbers. w = v - 3u, where v and u are any real numbers. I was told that I should set v and w to any arbitrary variable and then solve. Such as, let v = p and w = q, where p and q are any real numbers. I agree that this is one way to solve the system, but I don't see how my solution does not. Any suggestions?
 October 20th, 2013, 03:52 AM #2 Newbie   Joined: Dec 2012 Posts: 22 Thanks: 0 Re: Homogenous system with more variables than equations Edit: My answer also included x = 0 and y = 0.
 October 20th, 2013, 02:57 PM #3 Math Team     Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 234 Re: Homogenous system with more variables than equations You are writing u , v , and w in terms of each other so it looks 'circular'. Since v and w are free variables the convention is to rename them , this will give v = p w = q u = 1/3(p - q) x = 0 y = 0 Now , make arbitrary choices for p and q to get u , v , w , x , y to avoid the circular reasoning.
 October 20th, 2013, 04:58 PM #4 Newbie   Joined: Dec 2012 Posts: 22 Thanks: 0 Re: Homogenous system with more variables than equations Thanks, that makes sense now. Guess I have to admit it: I was wrong.
October 22nd, 2013, 02:27 AM   #5
Math Team

Joined: Jul 2011
From: North America, 42nd parallel

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Thanks: 234

Re: Homogenous system with more variables than equations

Quote:
 Originally Posted by nicnicman Thanks, that makes sense now. Guess I have to admit it: I was wrong.
Don't worry buddy , the first 4.5 billion years are tough ... thankfully they have passed already ... now it should be 'smooth sailing' from here on forward.

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