Parallel Lines — Cheat sheet

The condition

$m_1 = m_2$ with $n_1 \ne n_2$ - equal slopes, different intercepts. In standard form: $\dfrac{a_1}{a_2} = \dfrac{b_1}{b_2} \ne \dfrac{c_1}{c_2}$.

Building a parallel through a point

Shortcut

Parallel to $ax + by + c = 0$ is $ax + by + k = 0$: keep both coefficients, substitute the point for $k$.

Note

Two vertical lines are parallel although neither has a slope - the ratio test covers them, the slope test does not.