Show that two lines $a_{1}x + b_{1}y + c_{1} = 0$ and $a_{2}x + b_{2}y + c_{2} = 0$,where $b_{1}, b_{2} \neq 0$,are parallel if $\frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}}$.

(N/A) The given lines can be written in slope-intercept form $(y = mx + c)$ as follows:
$y = -\frac{a_{1}}{b_{1}}x - \frac{c_{1}}{b_{1}}$ $(1)$
$y = -\frac{a_{2}}{b_{2}}x - \frac{c_{2}}{b_{2}}$ $(2)$
The slopes of lines $(1)$ and $(2)$ are $m_{1} = -\frac{a_{1}}{b_{1}}$ and $m_{2} = -\frac{a_{2}}{b_{2}}$,respectively.
Two lines are parallel if and only if their slopes are equal,i.e.,$m_{1} = m_{2}$.
Substituting the values of the slopes,we get:
$-\frac{a_{1}}{b_{1}} = -\frac{a_{2}}{b_{2}}$
Multiplying both sides by $-1$,we obtain:
$\frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}}$
Thus,the lines are parallel if $\frac{a_{1}}{b_{1}} = \frac{a_{2}}{b_{2}}$.