Math101 · Independent Ontario learning library
Math101Point-Slope Form
Point-slope form $y-y_1=m(x-x_1)$ describes the line with slope $m$ through $(x_1,y_1)$. It comes from the slope relation $(y-y_1)/(x-x_1)=m$ and is especially useful when one point and slope are known.
Point-slope form translates local rate and one known state directly into a line. It supports modelling, tangent lines, and conversions among linear forms.
Intuition and core definition
Point-slope form $y-y_1=m(x-x_1)$ describes the line with slope $m$ through $(x_1,y_1)$. It comes from the slope relation $(y-y_1)/(x-x_1)=m$ and is especially useful when one point and slope are known.
Notation, language, and conditions
The paired coordinates must stay together: both subscripts refer to the same known point. Subtracting a negative coordinate creates addition. A vertical line has undefined slope and cannot be written in point-slope form; through $(a,b)$ it is $x=a$.
Why this idea matters
Point-slope form builds a line directly from one known point and a rate of change without first calculating the vertical intercept.
A dependable method
- Identify the given slope and one complete point.
- Substitute into $y-y_1=m(x-x_1)$ with parentheses around signed coordinates.
- Simplify double negatives but need not solve for $y$ unless another form is requested.
- Test the known point in the equation.
- Use the slope to generate a second point or convert forms for an additional check.
