Point-Slope Form of a Line — Mathematics Knowledge System

Overview

The point-slope form y - y1 = m(x - x1) expresses a line through a given point (x1, y1) with slope m. It is the most direct way to write a line equation when you know one point and the slope -- more natural than slope-intercept form when the y-intercept isn't immediately known.

Intuition

Slope = (y - y1)/(x - x1) for any point (x, y) on the line. Rearranging: y - y1 = m(x - x1). This is the point-slope form. Given two points, find the slope first, then plug into point-slope form using either point. Convert to slope-intercept by solving for y.

Formal Definition

Definition

A non-vertical line through point (x1, y1) with slope m has equation: y - y1 = m(x - x1). This is equivalent to y = m(x - x1) + y1, which gives y-intercept b = y1 - m*x1. Two points (x1,y1) and (x2,y2) determine slope m = (y2-y1)/(x2-x1), then the line is y - y1 = m(x - x1).

y - y_1 = m(x - x_1)

Point-slope form

m = \frac{y_2 - y_1}{x_2 - x_1}

Slope from two points

b = y_1 - mx_1

y-intercept from point and slope

Notation

Notation	Meaning
$(x_1, y_1)$	Given point on the line
$m$	Slope

Theorems

Theorem 1: Line Uniqueness

Through any point (x1,y1) with slope m there is exactly one (non-vertical) line, given by y - y1 = m(x - x1). Two distinct points determine a unique line.

Worked Examples

1
Plug into point-slope form.
$y - 5 = 3(x - 2)$
2
Expand to slope-intercept (optional): y = 3x - 6 + 5 = 3x - 1.
$y = 3x - 1$

✓ Answer

y - 5 = 3(x - 2), or equivalently y = 3x - 1.

Practice Problems

Easyfill in blank

Find the equation of the line through (0, 3) with slope -2.

Common Mistakes

Common Mistake

Writing y + y1 = m(x + x1) instead of y - y1 = m(x - x1).

The form has minus signs: the point (x1, y1) means we subtract x1 and y1. Watch signs when the given point has negative coordinates.

Quiz

The point-slope form y - y1 = m(x - x1) requires knowing:

The y-intercept and slopeOne point on the line and the slopeTwo points on the lineThe x-intercept and y-intercept

Historical Background

The equation of a line in various forms developed as coordinate geometry matured after Descartes (1637). The point-slope form follows directly from the definition of slope as rise/run, once you know that a unique line passes through any point with a given slope. It became standard in algebra curricula in the 20th century.

1637
Descartes introduces coordinate geometry in La Geometrie
Rene Descartes

Summary

Point-slope: y - y1 = m(x - x1) for line through (x1, y1) with slope m.
Two points -> find slope m = (y2-y1)/(x2-x1) -> plug into point-slope with either point.
Convert to slope-intercept by solving for y.

References

BookHall, B. and Fabricant, M. Algebra 1. Prentice Hall, 2001.