Equation of Straight Line in Plane/Point-Slope Form

Theorem
Let $\LL$ be a straight line embedded in a cartesian plane, given in slope-intercept form as:
 * $y = m x + c$

where $m$ is the slope of $\LL$.

Let $\LL$ pass through the point $\tuple {x_0, y_0}$.

Then $\LL$ can be expressed by the equation:
 * $y - y_0 = m \paren {x - x_0}$

Proof
As $\tuple {x_0, y_0}$ is on $\LL$, it follows that:

Substituting back into the equation for $\LL$:

Also presented as
This equation can also be seen presented as:


 * $y - y_0 = \paren {x - x_0} \tan \psi$

where $\psi$ is the angle that $\LL$ makes with the $x$-axis.