Determine whether the system of linear equations $2 x + 3 y - 5 = 0, 6 x + 9 y - 15 = 0$ has a unique solution, no solutions, or an infinite number of solutions.

Infinite number of solutions

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No solutions

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Unique solution

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Cannot be determined

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Solution

The correct option is A Infinite number of solutions
The equations are
$\Rightarrow 2 x + 3 y - 5 = 0$
here, $a_{1} = 2, b_{1} = 3, c_{1} = - 5$
$\Rightarrow 6 x + 9 y - 15 = 0$
here, $a_{2} = 6, b_{2} = 9, c_{2} = - 15$
$\Rightarrow \frac{a_{1}}{a_{2}} = \frac{2}{6} = \frac{1}{3}$
$\Rightarrow \frac{b_{1}}{b_{2}} = \frac{3}{9} = \frac{1}{3}$
$\Rightarrow \frac{c_{1}}{c_{2}} = \frac{- 5}{- 15} = \frac{1}{3}$
$\Rightarrow \frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}} = \frac{c_{1}}{c_{2}}$
Hence, the given system of equations has an infinite number of solutions.

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