Find the value of k for which the following lines intersect,
$6 x + 3 y = k - 3, 12 x + k y = k$

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Solution

$6 x + 3 y = k - 3, 12 x + k y = k$
It has polymony solution
$(6 x + 3 y = k - 3) \times 2 \Rightarrow 12 x + 6 y = 2 k - 6 - - - - (1)$
$12 x + k y = k - - - - (2)$
Fir the system to be polymony
$(1) = (2)$
$12 x + 6 y - 2 k + 6 = 12 x + k y - k$
On comparing
$k = 6, - 2 k + 6 = - k$
$\Rightarrow k = 6$
$∴$ the value of $k = 6$

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