Condition for Parallel Lines

Lines in the same plane that never intersect are parallel. For $y=m_1x+c_1$ and $y=m_2x+c_2$ to be parallel, the angle between them must be $0$ (or $\pi$): $\varphi=0\Rightarrow\tan\varphi=0\Rightarrow\dfrac{m_2-m_1}{1+m_1m_2}=0\Rightarrow m_2-m_1=0\Rightarrow m_2=m_1$. So parallel lines share the same slope; conversely, two non-vertical lines with equal slope are parallel. (All vertical lines — undefined slope — are trivially parallel to one another.)

In general form, $a_1x+b_1y+c_1=0$ and $a_2x+b_2y+c_2=0$ are parallel exactly when $\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}$, equivalently $a_1b_2=a_2b_1$.

Two useful ready-made results:

1. Every line parallel to $ax+by+c=0$ has the form $ax+by=k$ for some constant $k$ (only the constant term changes; the coefficients of $x,y$ — which fix the slope — are shared). …