# Equation of Straight Line in Plane/Point-Slope Form

## Theorem

Let $\mathcal L$ be a straight line embedded in a cartesian plane, given in slope-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:

 $\displaystyle y_0$ $=$ $\displaystyle m x_0 + c$ $\displaystyle \leadsto \ \$ $\displaystyle c$ $=$ $\displaystyle m x_0 - y_0$

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

 $\displaystyle y$ $=$ $\displaystyle m x + \paren {m x_0 - y_0}$ $\displaystyle \leadsto \ \$ $\displaystyle y - y_0$ $=$ $\displaystyle m \paren {x - x_0}$

$\blacksquare$