Equation of Straight Line in Plane/Point-Slope Form

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

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

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

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

Substituting back into the equation for $\mathcal L$: