Which of the following pairs of linear equations has unique solution, no solution, or infinitely many solutions. In case there is a unique solution, find it by using cross multiplication method.
(i) x – 3y – 3 = 0 and 3x – 9y – 2 = 0
(ii) 2x + y = 5 and 3x + 2y = 8
(iii) 3x – 5y = 20 and 6x – 10y = 40
(iv) x – 3y – 7 = 0 and 3x – 3y – 15 = 0

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Detailed Solution

(i) x – 3y – 3 = 0 and 3x – 9y – 2 = 0

$\frac{a_{1}}{a_{2}} = \frac{1}{3}, \frac{b_{1}}{b_{2}} = \frac{- 3}{- 9} = \frac{1}{3}, \frac{c_{1}}{c_{2}} = \frac{- 3}{- 2} = \frac{3}{2}$

$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$

Since, the given set of lines are parallel to each other, they will not intersect each other. Hence, there is no solution for these equations.

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