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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 $(-4,1)$ and $(8,4).$

Find the slope

$m=x_{2}−x_{1}y_{2}−y_{1} $

$m=-4−81−4 $

SubTermsSubtract terms

$m=-12-3 $

CalcQuotCalculate quotient

$m=0.25$

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

$y=0.25x+b$

$4=0.25⋅8+b$

MultiplyMultiply

$4=2+b$

SubEqn$LHS−2=RHS−2$

$2=b$

RearrangeEqnRearrange equation

$b=2$

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 $y=9x+4.$

Show 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 $y.$
The slope of the new line is $m=-1.$ Thus, we can write the complete equation as
$y=-x+4.$

$y=mx+4$

$1=m⋅3+4$

MultiplyMultiply

$1=3m+4$

SubEqn$LHS−4=RHS−4$

$-3=3m$

DivEqn$LHS/3=RHS/3$

$-1=m$

RearrangeEqnRearrange equation

$m=-1$

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 $m=runrise ,$ 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 $rise=6$ and the $run=2.$ Therefore, the slope is $m=26 =3.$

Replace $m$ with the slope

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