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?
$3 x - 5 y = 20, 6 x - 10 y = 40$

No solutions

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Unique solutions,

x = 0

and

y = 1

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Infinitely many solutions

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Data insufficient

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Solution

The correct option is B Infinitely many solutions
Given system of equations are:
$3 x - 5 y = 20$
$6 x - 10 y = 40$
Now rearranging both the equations, we get
$3 x - 5 y - 20$ ......(1)
$6 x - 10 y - 40$ ......(2)
From equation (1) and equation (2) will have
$a_{1} = 3, b_{1} = - 5, c_{1} = - 20$
$a_{2} = 6, b_{2} = - 10, c_{2} = - 40$
$\frac{a_{1}}{a_{2}} = \frac{3}{6} = \frac{1}{2}$
$\frac{b_{1}}{b_{2}} = \frac{- 5}{- 10} = \frac{1}{2}$
$\frac{c_{1}}{c_{2}} = \frac{- 20}{- 40} = \frac{1}{2}$
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
So, it will have infinity or many solutions.

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