Equation of a Straight Line

Three useful forms

For a non-vertical line, slope-intercept form is

Here $m$ is the slope and $(0,b)$ is the $y$-intercept. A line of slope $m$ through $(x_1,y_1)$ has point-slope form

A form that also includes vertical lines is

The last condition means $A$ and $B$ cannot both be zero. If $B\ne0$, rearrangement gives slope $-A/B$ and $y$-intercept $C/B$. If $B=0$, then $A\ne0$ and the line is vertical: $x=C/A$. These line representations are consistent with OpenStax’s treatment of linear functions.

Given one point and a slope

Find the line through $(2,5)$ with slope $-\tfrac32$. Insert the data into point-slope form:

Distribute and add 5:

To remove fractions, multiply the equation by 2 and rearrange:

All three equations describe the same line. The original point checks: $3(2)+2(5)=16$. The slope from standard form is $-3/2$, matching the given rate.

The geometric reasoning behind point-slope form is straightforward. Every other point on this non-vertical line has the same ratio $(y-y_1)/(x-x_1)=m$. Multiplying by $x-x_1$ produces the equation. The equation also includes the starting point itself, where both differences are zero, even though the ratio would be undefined there.

Given two distinct points

Find the line through $A=(-1,2)$ and $B=(3,10)$. First calculate the slope:

Use either point in point-slope form. Using $A$ gives

Check both points: at $x=-1$, the equation gives $y=2$; at $x=3$, it gives $y=10$. Checking both catches mistakes that a single check may miss.

When the points have the same horizontal coordinate

The points $(3,-2)$ and $(3,5)$ determine the vertical line $x=3$. Do not divide by zero while trying to force this into $y=mx+b$.

If the two supplied points are identical, they do not determine a unique line. A slope or another distinct point is needed. “Two points determine a line” assumes that the points are distinct.

Given intercepts

Suppose the intercepts are $(6,0)$ and $(0,4)$. Since both intercept values are non-zero, start with

Multiplying by 12 gives $2x+3y=12$. Solving for $y$ gives $y=-\tfrac23x+4$. The negative slope fits a line falling from $(0,4)$ towards $(6,0)$.

For a line through the origin, both axis intercepts coincide. Use another point or a slope instead of trying to put a zero intercept in a denominator.

Whole lines, segments and restricted models

The equation $y=2x+4$ describes an entire infinite line. To describe only the segment from $(-1,2)$ to $(3,10)$, include the restriction

The endpoint inequalities are part of the description. Without them, points beyond the segment still satisfy the equation. A vertical segment can instead be specified by a fixed $x$ and an interval for $y$.

A model may impose further restrictions. For example, a linear formula for the length of material remaining after cuts should not be extended to negative lengths just because the algebraic line continues.

Check more than appearance

Two drawings can look similar while representing different equations. A dependable check uses the exact slope and a known point. For two-point data, substitute both endpoints. For standard form, multiplying every coefficient and the constant by the same non-zero number produces an equivalent equation.

For instance, $3x+2y=16$ and $6x+4y=32$ are the same line. Multiplying only the left side changes the equation and generally changes the line. An equivalent line equation is a statement about the whole solution set, not just a cosmetic change of notation.

Related topics

Use Parallel and Perpendicular Lines to construct lines from a directional relationship. Distance from a Point to a Line uses standard form to measure the shortest separation.