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Linear function rules (linear equations) can be written in different ways to highlight different characteristics. In this section, writing these rules in slope-intercept form will be explored.

To write linear equations in slope-intercept form, $y=mx+b,$
the slope, $m,$ and the $y$-intercept, $b,$ of the line must be known. When two points on the line are known, the following method can be used.

Write the equation of the line that passes through the point $(\text{-} 4,1)$ and $(8,4).$

Find the slope

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

$m = \dfrac{{\color{#0000FF}{1}}-{\color{#009600}{4}}}{{\color{#0000FF}{\text{-} 4}}-{\color{#009600}{8}}}$

$m=\dfrac{\text{-} 3}{\text{-} 12}$

$m=0.25$

Replace $m$ with the slope

The equation $y=mx+b$ can be re-written with $m=0.25.$ This gives $y=0.25x+b.$

Find $b$ using a point

$y=0.25x+b$

${\color{#009600}{4}}=0.25\cdot {\color{#0000FF}{8}}+b$

$4=2+b$

$2=b$

$b=2$

Write the equation

Write the equation of the line that passes through the point $(3,1)$ and has the same $y$-intercept as the line $y=9x+4.$

A line in slope-intercept form is given by the equation
$y=mx+b,$
where $m$ is the slope of the line and $b$ is the $y$-intercept. The line $y=9x+4$ has a $y$-intercept of $(0,4).$ We want our line to have the same $y$-intercept. Therefore, the equation of the new line must also have the value $b=4.$ This gives $y=mx+4.$
Our line must also pass through the point $(3,1).$ We can solve for $m$ in the equation above by substituting this point for $x$ and $y.$
The slope of the new line is $m=\text{-} 1.$ Thus, we can write the complete equation as
$y=\text{-} x+4.$

$y=mx+4$

${\color{#009600}{1}}=m\cdot {\color{#0000FF}{3}} + 4$

$1= 3m + 4$

$\text{-} 3 = 3m$

$\text{-} 1 = m$

$m = \text{-} 1$

To write the equation of the graph of a line in slope-intercept form, $y=mx+b,$

the $y$-intercept, $b$, and the slope of the line, $m,$ must be found. The following method can be used. As an example, consider the line shown.

Find the $y$-intercept

Replace $b$ with the $y$-intercept

Find the slope

Next, the slope of the line must be determined. From a graph, the slope of a line can be expressed as $m=\dfrac{\text{rise}}{\text{run}},$ where rise gives the vertical distance between two points, and run gives the horizontal distance. To find the slope, use any two points. Here, use the already marked $y$-intercept and the arbitratily chosen $(2,2).$

From the lines drawn, it can be seen that the $\text{rise}=6$ and the $\text{run}=2.$ Therefore, the slope is $m=\dfrac{6}{2}=3.$

Replace $m$ with the slope

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