For a system of equations to be inconsistent, the two lines must be parallel, meaning they have the same slope but different constants.
Given equations:
$ \displaystyle 3x - 7y = 1 $
$ \displaystyle kx + 14y = 6 $
Rewrite in the form $ \displaystyle a_1x + b_1y + c_1 = 0 $ and $ \displaystyle a_2x + b_2y + c_2 = 0 $:
$ \displaystyle a_1 = 3, \quad b_1 = -7, \quad c_1 = 1 $
$ \displaystyle a_2 = k, \quad b_2 = 14, \quad c_2 = 6 $
For inconsistency, the condition is:
$ \displaystyle \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} $
Substituting values:
$ \displaystyle \frac{3}{k} = \frac{-7}{14} $
$ \displaystyle \frac{3}{k} = -\frac{1}{2} $
Cross multiply:
$ \displaystyle 3 \times 2 = -1 \times k $
$ \displaystyle k = -6 $
Thus, the value of $ \displaystyle k $ for which the system is inconsistent is $ \displaystyle -6 $.