Equation of Straight Line in Plane/Point-Slope Form

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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 $\mathcal L$:

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

$\blacksquare$


Sources