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November 3rd, 2014, 09:29 AM  #1 
Newbie Joined: Nov 2014 From: turkey Posts: 9 Thanks: 0  Systems of Linear Equations(Immediate)
Suppose that a system of linear equation in four variables has ten equations. Does it mean that the solution set cannot have free variables? 
November 3rd, 2014, 09:36 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,690 Thanks: 2669 Math Focus: Mainly analysis and algebra 
It depends. If the equations are inconsistent, there are zero solutions. If the equations are consistent and there is a set of four of them that are linearly independant, there will be a unique solution with no free variables. If the equations are consistent and there is a set of three of them that are linearly independant (but no such set of four), there will be a solution having one free variable. If the equations are consistent and there is a set of two of them that are linearly independant (but no such set of three or four), there will be a solution with two free variables. If the equations are consistent but none are linearly independant (i.e. all the equations are equivalent), there will be a solution having three free variables. 

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