Without actually solving the simultaneous equations given below,
decide which simultaneous equations have unique solutions ,
no solution or infinitely many solutions.
$(ii) 3 y = 2 - x; 3 x = 6 - 9 y$

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Solution

$3 y = 2 - x 3 x = 6 - 9 y$

These equations can be written as follows:

$x + 3 y = 2 3 x + 9 y = 6$

Comparing the above equations with the equations $a_{1} x + b_{1} y = c_{1} and a_{2} x + b_{2} y = c_{2}, we get$ :

$a_{1} = 1 b_{1} = 3 c_{1} = 2 a_{2} = 3 b_{2} = 9 c_{2} = 6 ∴ \frac{a_{1}}{a_{2}} = \frac{1}{3}; \frac{b_{1}}{b_{2}} = \frac{3}{9} = \frac{1}{3} and \frac{c_{1}}{c_{2}} = \frac{2}{6} = \frac{1}{3}$

$i . e ., \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} ∴ The given equations have infinitely many solutions .$

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