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August 29th, 2016, 03:46 PM  #1 
Member Joined: Mar 2016 From: NJ Posts: 48 Thanks: 2  echelon method expressing with infinite solutions
Hi, Can someone explain to me how you express the answer for using the echelon method when there are an infinite number of solutions? For example, if you have a matrix of: 1 2 l 3 0 0 l 0 How do I express the answer? For my homework, it has to be ( , y) but I have no idea what goes in the first part and there is not a single example in my book of how to express it. Last edited by blimper; August 29th, 2016 at 03:54 PM. 
August 29th, 2016, 04:06 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,382 Thanks: 1281 
$(1,2)\cdot (x,y) = 3$ $x + 2y = 3$ $y = \dfrac{3x}{2}$ so all vectors $c\begin{pmatrix}x \\ \dfrac {3x}{2}\end{pmatrix},~c \in \mathbb{R}$ are solutions 
August 29th, 2016, 04:31 PM  #3 
Member Joined: Mar 2016 From: NJ Posts: 48 Thanks: 2 
I guess what i was thinking is if I had something like: 1 5/6 l 1/6 0 0 l 0 It would become: x5/6y= 1/6 leading to x = 5y1/6 Where my answer would be 5y1 ( , y ) 6 (Is there a tutorial for how to express things on here like you did?) 
August 29th, 2016, 04:48 PM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 2,382 Thanks: 1281  Quote:
$x  \dfrac 5 6 y = \dfrac 1 6$ $6x 5y = 1$ $y = \dfrac {6x1}{5}$ so all vectors $c \begin{pmatrix} x \\ \frac{6x1}{5}\end{pmatrix},~c \in \mathbb{R}$  
August 30th, 2016, 06:53 PM  #5 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 
1 2 1 3 0 0 1 0 is the matrix of the system of equations x+2y+z=3 0x+0y+z=0 in augmented form. In rref, by elementary row operations which don't change the solution, the matrix is 1 2 0 3 0 0 1 0 from which z=0, x+2y=3 y is arbitrary and x=32y 

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