Do the following equations represent a pair of coincident lines? Justify your answer.

(i) and 7x + 3y = 7

(ii) - 2x - 3y = 1 and 6y + 4x = - 2

(iii)

Condition for coincident lines,

a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2};

(i) No.

Given pair of linear equations

and 7x + 3y = 7

Comparing with ax + by + c = 0;

Here, a_{1} = 3, b_{1} = 1/7, c_{1} = - 3;

And a_{2} = 7, b_{2} = 3, c_{2} = - 7;

a_{1} /a_{2} = 3/7

b_{1} /b_{2} = 1/21

c_{1} /c_{2} = - 3/ - 7 = 3/7

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

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

(ii) Yes, given pair of linear equations.

- 2x - 3y - 1 = 0 and 4x + 6y + 2 = 0;

Comparing with ax + by + c = 0;

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

And a_{2} = 4, b_{2} = 6, c_{2} = 2;

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

b_{1} /b_{2} = - 3/6 = - 1/2

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

Here, a_{1}/a_{2} = b_{1}/b_{2} = c_{1}/c_{2}, i.e. coincident lines

Hence, the given pair of linear equations is coincident.

(ii) No, given pair of linear equations are

and = 0

Comparing with ax + by + c = 0;

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

And a_{2} = 4, b_{2} = 8, c_{2} = 5/16;

a_{1} /a_{2} = 1/8

b_{1} /b_{2} = 1/8

c_{1} /c_{2} = 32/25

Here, a_{1}/a_{2} = b_{1}/b_{2} c_{1}/c_{2}, i.e. parallel lines

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

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