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# Writing Linear Equations in Slope-Intercept Form

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.
Method

## Writing the Equation of a Line in Slope-Intercept Form using Two Points

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 $(8,4).$

### 1

Find the slope
When two points on a line are known, the slope of the line can be found using the slope formula. Here, the coordinates $(8,4)$ and $(\text{-} 4,1)$ can be substituted in place of $(x_1,y_1)$ and $(x_2,y_2),$ respectively.
$m = \dfrac{y_2-y_1}{x_2-x_1}$
$m = \dfrac{{\color{#0000FF}{1}}-{\color{#009600}{4}}}{{\color{#0000FF}{\text{-} 4}}-{\color{#009600}{8}}}$
$m=\dfrac{\text{-} 3}{\text{-} 12}$
$m=0.25$
The slope of the line passing through the two points is $m=0.25.$

### 2

Replace $m$ with the slope

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

### 3

Find $b$ using a point
Next, the $y$-intercept can be found by replacing $x$ and $y$ in the equation with either of the given points. Then, solving for $b$ gives the $y$-intercept. Here, the arbitrarily chosen point that will be used is $(8,4).$ Therefore, substitute $x=8$ and $y=4$ into the equation from Step 2.
$y=0.25x+b$
${\color{#009600}{4}}=0.25\cdot {\color{#0000FF}{8}}+b$
$4=2+b$
$2=b$
$b=2$
Thus, the $y$-intercept is $b=2.$

### 4

Write the equation

Lastly, the complete equation in slope-intercept form can be written by replacing the value of the $y$-intercept found above. Here, $b=2$ will be substituted into the equation from Step 2. This gives $y=0.25x+2.$

Exercise

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

Solution
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 $y.$
$y=mx+4$
${\color{#009600}{1}}=m\cdot {\color{#0000FF}{3}} + 4$
$1= 3m + 4$
$\text{-} 3 = 3m$
$\text{-} 1 = m$
$m = \text{-} 1$
The slope of the new line is $m=\text{-} 1.$ Thus, we can write the complete equation as $y=\text{-} x+4.$
info Show solution Show solution
Method

## Writing the Equation of a Line in Slope-Intercept Form from a Graph

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.

### 1

Find the $y$-intercept

The $y$-intercept is the point where the graph intersects the $y$-axis. From the diagram, it can be seen that the $y$-intercept is $(0,\text{-} 4).$

### 2

Replace $b$ with the $y$-intercept

The $y$-coordinate of the $y$-intercept can be substituted into $y=mx+b$ for $b.$ Here, substituting $b=\text{-} 4$ gives $y=mx-4.$

### 3

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 $m=\dfrac{6}{2}=3.$

### 4

Replace $m$ with the slope

The complete equation of the line can now be written by substituting the value of $m$ into the equation from Step 2. Here, substitute $m=3.$ $y=3x-4.$

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