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Consider a non- homogeneous system of linear equations representing mathematically an over determined system. Such a system will be

(A) consistent having a unique solution

(B) consistent having many solutions

(C) inconsistent having a unique solution

(D) inconsistent having no solution

3 Answers

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A system of linear equations with more equations than unknowns is sometimes called an overdetermined system. 

A system of linear equations with more equations than unknowns can be consistent with unique or infinite solutions as well as it could be inconsistent because to make number of equations more than the number of unknowns we can always add any equation which is  linear combination of existing set of equations.  


A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system.  

An underdetermined system always has more variables than equations. There cannot be more basic variables than there are equations, so there must be at least one free variable. Such a variable may be assigned infinitely many different values. If the system is consistent, each different value of a free variable will produce a different solution. So, An underdetermined system can NEVER be "Consistent with Unique Solution".

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Answer is : (D) inconsistent having no solution

 A system of equations is considered overdetermined if there are more equations than unknowns.

Different cases of a overdetermined system , which are inconsistent and have no solution: 

case 1: Consider the system of 3 equations and 2 unknowns (X and Y), which is overdetermined because 3>2,

{\displaystyle {\begin{aligned}Y&=-2X-1\\Y&=3X-2\\Y&=X+1.\end{aligned}}}

There is one solution for each pair of linear equations: for the first and second equations (0.2, −1.4), for the first and third (−2/3, 1/3), and for the second and third (1.5, 2.5). However, there is no solution that satisfies all three simultaneously.

case 2:    A system of three linearly independent equations, three lines (two parallel), no solutions.

                                                                                                                            

case 3:  A system of three linearly independent equations, three lines (all parallel), no solution

In systems of linear equations, Li=ci for 1 ≤ i ≤ M, in variables X1X2, ..., XN the equations are sometimes linearly dependent; in fact the number of linearly independent equations cannot exceed N+1. We have the following possible cases for an overdetermined system with N unknowns and M equations (M>N).

  • M = N+1 and all M equations are linearly independent. This hull yields no solution. Example: x = 1, x = 2.
  • M > N but only K equations (K < M and K ≤ N+1) are linearly independent. There exist three possible sub-cases of this:
    • K = N+1. This hull yields no solutions. Example: 2x = 2, x = 1, x = 2.
    • K = N. This hull yields either a single solution or no solution, the latter occurring when the coefficient vector of one equation can be replicated by a weighted sum of the coefficient vectors of the other equations but that weighted sum applied to the constant terms of the other equations does not replicate the one equation's constant term. Example with one solution: 2x = 2, x = 1. Example with no solution: 2x + 2y = 2, x + y = 1, x + y = 3.
    • K < N. This hull yields either infinitely many solutions or no solution, the latter occurring as in the previous sub-. Example with infinitely many solutions: 3x + 3y = 3, 2x + 2y = 2, x + y = 1. Example with no solution: 3x + 3y + 3z = 3, 2x + 2y + 2z = 2, x + y + z = 1, x + y + z = 4.
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Answer is a), b) and d)

In, Overdetermined system number of equations > number of unknowns.

So overdetermined system in most cases will be inconsistent(no solutions) for equations having random coefficients.

However Overdetermined system will have solutions in some cases.

    Case (i) :- if some equation occurs several times in the system.

                       eg(i) :- x + 2y = 2; x + 4y = 3; x + 4y = 3 --------> This system of equations are consistent with unique solution

Case (ii) :-  if some equations are linear combinations of the others.

                   eg(ii) :- x + y =3;  3x + 3y = 9 ; 5 x + 5y = 15  --------> This system of equations are consistent with infinite number of solutions.

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