For what value of k do the set of equations $4 x - (3 k + 2) y = 20$ and $(11 k - 3) x - 10 y = 40$ have infinite solutions ?

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Solution

Given equations are
$4 x - (3 k + 2) y = 20$
$(11 k - 3) x - 10 y = 40$
These equations have infinite solutions if, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$
$\frac{4}{11 k - 3} = \frac{- (3 k + 2)}{- 10} = \frac{- 20}{- 40}$
$\frac{4}{11 k - 3} = \frac{1}{2}$
$8 = 11 k - 3$
$11 k = 11$
$k = 1$

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