Skill: Write the equation of a line through two points

To write the equation of a line through two points:

1. Find the slope:

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

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

$$y-y_1=m(x-x_1)$$

3. Simplify if desired into slope-intercept form:

$$y=mx+b$$

4. 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 $x=c$.