October 15th, 2018, 05:09 AM  #1 
Newbie Joined: Jan 2014 Posts: 19 Thanks: 0  Linear system
Question: Using the fact that $\displaystyle \textbf{Ax=b}$ is consistent if and only if $\displaystyle \textbf{b}$ is a linear combination of the columns of $\displaystyle \textbf{A}$ to find a solution to $\displaystyle \left( \begin{array}{cccc} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \\ 3 & 4 & 1 & 2 \end{array} \right)$$\displaystyle \left( \begin{array}{c} x\\ y\\ z\\ w \end{array} \right)$$\displaystyle =\left( \begin{array}{c} 20\\ 20\\ 20\end{array} \right).$ My work: I rewrite the given system into $\displaystyle x\left( \begin{array}{c} 1\\ 2\\ 3\end{array} \right)$$\displaystyle +y\left( \begin{array}{c} 2\\ 3\\ 4\end{array} \right)$$\displaystyle +z\left( \begin{array}{c} 3\\ 4\\ 1\end{array} \right)$$\displaystyle +w\left( \begin{array}{c} 4\\ 1\\ 2\end{array} \right)$$\displaystyle =\left( \begin{array}{c} 20\\ 20\\ 20\end{array} \right).$ What should I do next? Please give some hints. Thank you. 
October 15th, 2018, 08:51 AM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 502 Thanks: 280 Math Focus: Dynamical systems, analytic function theory, numerics 
Notice that all 3 of the rows sums to 10.

October 15th, 2018, 09:15 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,868 Thanks: 1833 
As, x = y = z = w = 2 is an obvious solution, the system is consistent. You can use $\text{uAx = ub}$, where $\text{u}$ is the vector (1, 2, 1) if you want to find out more. 
October 16th, 2018, 08:35 AM  #4  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote:
x+3y+4z+ w=20 3x+4y+z+2w=20 Put in augmented matrix form and sove by row reduction to get: 1 2 3  20 4w 2 3 4  20 w 3 4 1  20 2w 1 2 3  20 4w 0 1 2  20 7w 0 0 2 0 2w z=w y=2011w x=209w $\displaystyle \begin{bmatrix} x\\ y\\ z \end{bmatrix}=\begin{bmatrix} 209w\\ 2011w\\ w \end{bmatrix}=\begin{bmatrix} 20\\ 20\\ 0 \end{bmatrix}+w\begin{bmatrix} 9\\ 11\\ 1 \end{bmatrix} $ You can ck the algebra.  

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algebra or trigonometry, linear, linear algebra, linear system equations, matrices, precalculus, system 
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