Skill: Write equations of lines in slope-intercept form

A line in slope-intercept form is written as

$$y=mx+b$$

where:

• $m$ is the slope, or rate of change
• $b$ is the $y$-intercept, the point where the line crosses the $y$-axis

Example

Write the equation of the line passing through $(2,5)$ and $(6,13)$.

Step 1: Find the slope.

Use

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

Substitute the two points:

$$m=\frac{13-5}{6-2}=\frac{8}{4}=2$$

So the equation begins as

$$y=2x+b$$

Step 2: Find $b$ using one point.

Use $(2,5)$, so $x=2$ and $y=5$:

$$5=2(2)+b$$

Solve for $b$:

$$5=4+b$$ $$b=1$$

Step 3: Write the equation.

Substitute $m=2$ and $b=1$ into $y=mx+b$:

$$\boxed{y=2x+1}$$

Step 4: Check the other point.

For $(6,13)$:

$$y=2(6)+1=13$$

The equation works for both points. Notice that the $y$-intercept is $(0,1)$, not either of the given points.