October 15th, 2018, 05:09 AM  #1 
Newbie Joined: Jan 2014 Posts: 20 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: 680 Thanks: 454 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: 21,105 Thanks: 2324 
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  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  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.  

Tags 
algebra or trigonometry, linear, linear algebra, linear system equations, matrices, precalculus, system 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Nonlinear system  Jhenrique  Linear Algebra  2  June 15th, 2015 03:41 AM 
System of Linear Equation  App  Algebra  2  February 21st, 2015 03:09 PM 
Nonlinear system of equations  mared  Algebra  2  February 18th, 2015 01:21 PM 
Linear approximation of Non linear system by Taylor series  RGNIT  Calculus  0  March 21st, 2014 12:21 AM 
Solve nonlinear system (=linear+linear systems)  tools  Linear Algebra  1  September 21st, 2012 01:38 PM 