$A \left(7, -5\right)$, $B \left(5,3\right)$ and $C \left(-9, 1\right)$ are the vertices of $\triangle ABC$. Find the slope of $BC$ and the equation of the altitude through $A$.
Given: $\;$ $A \left(7, -5\right)$, $B \left(5,3\right)$, $C \left(-9, 1\right)$
Slope of $BC = m_1 = \dfrac{3 - 1}{5 + 9} = \dfrac{2}{14} = \dfrac{1}{7}$
The altitude through $A$ is perpendicular to $BC$.
$\therefore \;$ Slope of altitude through $A = m = \dfrac{-1}{m_1} = -7$
The altitude passes through $A \left(7, -5\right)$
$\therefore \;$ The required equation of altitude is
$y + 5 = -7 \left(x - 7\right)$
i.e. $\;$ $y + 5 = -7x + 49$
i.e. $\;$ $7x + y = 44$