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Concept

Point-Slope Form

The equation of a line can be written in slope-intercept form or standard form. However, when the slope and a point are given, point-slope form is another alternative.

$y - y_{1} = m (x - x_{1})$

Here, $m$ is the slope and $(x_{1}, y_{1})$ is a point on the line. By simplifying and solving for $y,$ it's possible to rewrite a point-slope equation in slope-intercept form: $y = m x + b .$

Derivation

y - y_{1} = m (x - x_{1})

Point-slope form is a manipulated version of the slope formula.

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

MultEqn

$LHS \cdot (x_{2} - x_{1}) = RHS \cdot (x_{2} - x_{1})$

m (x_{2} - x_{1}) = y_{2} - y_{1}

RearrangeEqn

Rearrange equation

y_{2} - y_{1} = m (x_{2} - x_{1})

The specific point

(x_{2}, y_{2})

is then replaced by the more general

(x, y)

, which gives the point-slope form.

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Concept

Point-Slope Form

Derivation

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