Skill: Write a linear equation in slope-intercept form

A linear equation in slope-intercept form is

$$y=mx+b$$

where:

• $m$ is the slope, or constant rate of change
• $b$ is the $y$-intercept, the value of $y$ when $x=0$

Example

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

Step 1: Find the slope.

Use

$$m=\frac{\text{change in }y}{\text{change in }x} =\frac{y_2-y_1}{x_2-x_1}$$

Substitute the two points:

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

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

Step 2: Start with slope-intercept form.

$$y=mx+b$$

Substitute $m=2$:

$$y=2x+b$$

Step 3: Find the $y$-intercept.

Use either given point. Substitute $(2,5)$:

$$5=2(2)+b$$ $$5=4+b$$ $$b=1$$

So the line crosses the $y$-axis at $(0,1)$.

Step 4: Write the equation.

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

Check: If $x=6$, then

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

This matches the second point, so the equation is correct.