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

a_{1}/a_{2} b_{1}/b_{2.} [unique solution]

and a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2} [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, a_{1} = - 3, b_{1} = - 4, c_{1} = - 12;

And a_{2} = 3, b_{2} = 4, c_{2} = - 12;

a_{1} /a_{2} = - 3/3 = - 1

b_{1} /b_{2} = - 4/4 = - 1

c_{1} /c_{2} = - 12/ - 12 = 1

Here, a_{1}/a_{2} = b_{1}/b_{2} c_{1}/c_{2}

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, a_{1} = 3/5, b_{1} = - 1, c_{1} = - 1/2;

And a_{2} = 1/5, b_{2} = 3, c_{2} = - 1/6;

a_{1} /a_{2} = 3

b_{1} /b_{2} = - 1/ - 3 = 1/3

c_{1} /c_{2} = 3

Here, a_{1}/a_{2} b_{1}/b_{2.}

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, a_{1} = 2a, b_{1} = b, c_{1} = - a;

And a_{2} = 4a, b_{2} = 2b, c_{2} = - 2a;

a_{1} /a_{2} = 1/2

b_{1} /b_{2} = 1/2

c_{1} /c_{2} = 1/2

Here, a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}

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, a_{1} = 1, b_{1} = 3, c_{1} = 11

And a_{2} = 2, b_{2} = 6, c_{2} = 11

a_{1} /a_{2} = 1/2

b_{1} /b_{2} = 1/2

c_{1} /c_{2} = 1

Here, a_{1}/a_{2} = b_{1}/b_{2} c_{1}/c_{2}.

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

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