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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, 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. For example, to write the equation of the line that passes through the points (-4,1) and (8,4), there are four steps to follow.
### 1

Given two points on a line, the slope of the line can be found by using the Slope Formula. In this case, the coordinates (8,4) and (-4,1) can be substituted in place of (x1,y1) and (x2,y2), respectively.
The slope m of the line passing through the two points is 0.25.
### 2

Now that the value of the slope is known, it can be substituted for m in the slope-intercept form of an equation.
### 3

Next, the y-intercept can be found by substituting either of the given points into the equation. Then, solving for b results in finding the y-intercept. In the considered example, (8,4) can be used. Substitute its coordinates into the equation from Step 2.
Therefore, the y-intercept is 2.
### 4

Lastly, the complete equation in slope-intercept form can be written by substituting the value of the y-intercept found above into the equation from Step 2.

Find the slope

$m=x_{2}−x_{1}y_{2}−y_{1} $

SubstitutePoints

Substitute $(-4,1)$ & $(8,4)$

$m=8−(-4)4−1 $

SubNeg

a−(-b)=a+b

$m=8+44−1 $

AddSubTerms

Add and subtract terms

$m=123 $

CalcQuot

Calculate quotient

m=0.25

Replace m with the slope

Find b using a point

y=0.25x+b

SubstituteII

x=8, y=4

4=0.25(8)+b

Multiply

Multiply

4=2+b

SubEqn

LHS−2=RHS−2

2=b

RearrangeEqn

Rearrange 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
The slope of the new line is m=-1. Thus, we can write the complete equation as

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

SubstituteII

x=3, y=1

1=m⋅3+4

Multiply

Multiply

1=3m+4

SubEqn

LHS−4=RHS−4

-3=3m

DivEqn

$LHS/3=RHS/3$

-1=m

RearrangeEqn

Rearrange equation

m=-1

y=-x+4.

To write the equation of the graph of a line in slope-intercept form, ### 1

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,-4).

### 2

The y-coordinate of the y-intercept can be substituted into y=mx+b for b. Here, substituting b=-4 gives
### 3

Next, the slope of the line must be determined. From a graph, the slope of a line can be expressed as
### 4

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=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

y=mx−4.

Find the slope

$m=runrise ,$

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
Replace m with the slope

y=3x−4.

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