1. Consider the systems, each consisting of

**m**linear equations in

**n**variables.

- if m < n, then all such systems have a solutions.
- If m>n, then non of these systems has a solutions.
- if m=n, then there exits a system which has a solution.

**CORRECT**?

- i,ii and iii are true
- Only ii and iii are true
- Only iii is true
- None of them is true.

Solution:

__Statement i__
Consider 2 equations in three variables.

i.e., m=2 & n=3 (m<n)

x - y + z = 1

-x + y -z = 2

This system has no solutions is inconsistent (x=1 and y=1).

i.e.,

*"For ‘n’ variables, we need atleast ‘n’ linear equations to find value of each variable"*

**∴ Statement i is false.**

__Statement ii__
i.e., m=3 & n=2 (m>n)

x + y = 2,

x - y = 0,

3x + y = 4

This system has a unique solutions (no solution)

unique solutions :

unique solutions :

*"A system of linear equations ax + by + c = 0 and dx + ey + g = 0 will have a unique solution if the two lines represented by the equations ax + by + c = 0 and dx + ey + g = 0 intersect at a point. i.e., if the two lines are neither parallel nor coincident. Essentially, the slopes of the two lines should be different."*

*∴ Statement ii is false.*

__Statement iii.__
Consider a system with 2 equations and two variables.

i.e, m=2 & n=2 (m=n)

x + y = 2 and x - y =0

The system has a solution x = 1 and y = 1

*∴ Statement iii is true.*

*Hence ANSWER is c*
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