Write the equation of a line through two points

Given two distinct points in the Cartesian plane, the learner determines the line’s slope as the change in $y$ divided by the change in $x$ , then uses that relationship to write and interpret an equation in point-slope, slope-intercept, or an equivalent form. This includes recognizing horizontal lines as having zero slope and vertical lines as having undefined slope, and checking that both points satisfy the equation; parametric, vector, three-dimensional, and other advanced line representations are not included.

Detailed Explanation: Write the equation of a line through two points

To write the equation of a line through two points:

Find the slope:

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

Use the slope and either point in point-slope form:

y-y_1=m(x-x_1)

Simplify if desired into slope-intercept form:

y=mx+b

Check that both points satisfy the equation.

Example

Write the equation of the line through $(2,3)$ and $(6,11)$ .

Step 1: Find the slope.

Let $(x_1,y_1)=(2,3)$ and $(x_2,y_2)=(6,11)$ .

m=\frac{11-3}{6-2} =\frac{8}{4} =2

The slope is $2$ , meaning $y$ increases by $2$ for every increase of $1$ in $x$ .

Step 2: Use point-slope form.

Use the point $(2,3)$ :

y-3=2(x-2)

This is a correct equation of the line.

Step 3: Rewrite in slope-intercept form.

Distribute and simplify:

y-3=2x-4

y=2x-1

So the equation is

\boxed{y=2x-1}

Step 4: Check both points.

For $(2,3)$ :

y=2(2)-1=3

For $(6,11)$ :

y=2(6)-1=11

Both points satisfy the equation, so the equation is correct.

Remember: A horizontal line has slope $0$ and can be written as $y=c$ . A vertical line has an undefined slope and can be written as $x=c$ .

Learn by doing: Write the equation of a line through two points

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Equation of a Line from Coordinates of Points

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