General definition of System of Linear Equations says that

"If The system has a unique solution, It has independent set of Equations"

Consider the system of linear equations $$x-2y=-1$$ $$3x+5y=8$$ $$4x+3y=7$$ As we can see from the below graph that all the 3 line intersect at a single point $\implies$ System has a unique solution. But at the same time system is not independent as any equation can be derived from the algebraic manipulations of other two equations. So, how definition is true.


  • 1
    $\begingroup$ What do you mean by an independent set of equations? By independent they could mean that the columns of the associated matrix are independent vectors. $\endgroup$
    – Wintermute
    Dec 30 '14 at 18:25
  • $\begingroup$ @mtiano From independent set of equations I meant, No equation can be obtained from algebraic manipulation of other two equations. $\endgroup$
    – Atinesh
    Dec 30 '14 at 18:32
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    $\begingroup$ The statement is missing the qualifier that there should be the same number of equations and variables. $\endgroup$ Dec 30 '14 at 18:35
  • $\begingroup$ The system under consideration is an overdetermined system that, in this case, has a unique solution because it contains sufficient dependent equations such that the number of independent equations is exactly equal to the number of unknowns, which is the missing condition in the definition. $\endgroup$
    – PYK
    Dec 30 '14 at 18:56
  • $\begingroup$ @PYK Now I got it. I'd a misconception, You are right. $\endgroup$
    – Atinesh
    Dec 31 '14 at 10:20

Above System of equations are an example of Overdetermined System

A system of equations is considered overdetermined, If there are more equations than unknowns. The only cases where the overdetermined system will have a solution is when it contains enough linearly dependent equations that the number of independent equations does not exceed the number of unknowns [Wiki].


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