Write the number of solutions of the following pair of linear equations:

$x + 2 y - 8 = 0, 2 x + 4 y = 16.$

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Solution

Coefficients of the equation $x + 2 y - 8 = 0$ are $a_{1} = 1, b_{1} = 2$ and $c_{1} = - 8$
$2 x + 4 y = 16 \Rightarrow 2 x + 4 y - 16 = 0$
Coefficients of the equation $2 x + 4 y - 16 = 0$ are $a_{2} = 2, b_{2} = 4$ and $c_{2} = - 16$

$\frac{a_{1}}{a_{2}} = \frac{1}{2}, \frac{b_{1}}{b_{2}} = \frac{2}{4} = \frac{1}{2}, \frac{c_{1}}{c_{2}} = \frac{- 8}{- 16} = \frac{1}{2} .$

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

Hence, there are infinitely many solutions to the pair of linear equations.

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