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Prove that the system of equations given as $2x-3y = 7$ and $4x + ky = 9$, is inconsistent for $k = -6$. Also, obtain the solution of the system of equations, if $k = -1$.
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Solution: (a) $\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$, $\frac{b_1}{b_2} = \frac{-3}{k}$, $\frac{c_1}{c_2} = \frac{7}{9}$
For $k = -6$, $\frac{b_1}{b_2} = \frac{-3}{-6} = \frac{1}{2}$ (1/2 Mark)
$:: \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
$:::$ system of equations is inconsistent (1/2 Mark)
For $k = -1$, the system of equations is
$2x - 3y = 7$
$4x - y = 9$
solving, we get $x = 2$ and $y = -1$ (1/2+1/2 Mark)
For $k = -6$, $\frac{b_1}{b_2} = \frac{-3}{-6} = \frac{1}{2}$ (1/2 Mark)
$:: \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$
$:::$ system of equations is inconsistent (1/2 Mark)
For $k = -1$, the system of equations is
$2x - 3y = 7$
$4x - y = 9$
solving, we get $x = 2$ and $y = -1$ (1/2+1/2 Mark)