Are the following pair of linear equations consistent? Justify your answer.

(i) - 3x - 4y = 12 and 4y + 3x = 12


(ii)


(iii) 2ax + by = a and 4ax + 2by - 2a = 0; a, b 0


(iv) x + 3y = 11 and 2x + 6y = 11


Conditions for pair of linear equations are consistent


a1/a2 b1/b2. [unique solution]


and a1/a2 = b1/b2 = c1/c2 [coincident or infinitely many solutions]


(i) No.


The given pair of linear equations -


- 3x - 4y - 12 = 0 and 4y + 3x - 12 = 0


Comparing with ax + by + c = 0;


Here, a1 = - 3, b1 = - 4, c1 = - 12;


And a2 = 3, b2 = 4, c2 = - 12;


a1 /a2 = - 3/3 = - 1


b1 /b2 = - 4/4 = - 1


c1 /c2 = - 12/ - 12 = 1


Here, a1/a2 = b1/b2 c1/c2


Hence, the pair of linear equations has no solution, i.e., inconsistent.


(ii) Yes.


The given pair of linear equations


and


Comparing with ax + by + c = 0;


Here, a1 = 3/5, b1 = - 1, c1 = - 1/2;


And a2 = 1/5, b2 = 3, c2 = - 1/6;


a1 /a2 = 3


b1 /b2 = - 1/ - 3 = 1/3


c1 /c2 = 3


Here, a1/a2 b1/b2.


Hence, the given pair of linear equations has unique solution, i.e., consistent.


(iii) Yes.


The given pair of linear equations -


2ax + by –a = 0 and 4ax + 2by - 2a = 0


Comparing with ax + by + c = 0;


Here, a1 = 2a, b1 = b, c1 = - a;


And a2 = 4a, b2 = 2b, c2 = - 2a;


a1 /a2 = 1/2


b1 /b2 = 1/2


c1 /c2 = 1/2


Here, a1/a2 = b1/b2 = c1/c2


Hence, the given pair of linear equations has infinitely many solutions, i.e., consistent or dependent.


(iv) No.


The given pair of linear equations


x + 3y = 11 and 2x + 6y = 11


Comparing with ax + by + c = 0;


Here, a1 = 1, b1 = 3, c1 = 11


And a2 = 2, b2 = 6, c2 = 11


a1 /a2 = 1/2


b1 /b2 = 1/2


c1 /c2 = 1


Here, a1/a2 = b1/b2 c1/c2.


Hence, the given pair of linear equations has no solution.


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