Find k for which the system x + 2y = 3 and 5x + ky + 7 = 0 is inconsistent.

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Solution

The given system is
$x + 2 y - 3 = 0 . . . . . (i) 5 x + k y + 7 = 0 . . . . . (i i)$
Here, $a_{1} = 1, b_{1} = 2, c_{1} = - 3, a_{2} = 5, b_{2} = k and c_{2} = 7$ .
For the system to be inconsistent, we must have
$\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} \Rightarrow \frac{1}{5} = \frac{2}{k} \neq \frac{- 3}{7} \Rightarrow \frac{1}{5} = \frac{2}{k} \Rightarrow k = 10$
Hence, k = 10.

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5 x + k y = - 7, x + 2 y = 3

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k

x + 2 y = 3

5 x + k y + 7 = 0

k

x + 2 y - 3 = 0

5 x + k y + 7 = 0

x + 2 y = 3 and 5 x + k y + 7 = 0

k \neq 10

k = \frac{- 7}{3}

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