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Writing Linear Equations

Writing Linear Equations in Slope-Intercept Form

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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.

Method

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

To write linear equations in slope-intercept form, the slope m and the y-intercept b of the line must be known.
When only two points on the line are known, the following four-step method can be used. For example, the equation of the line that passes through the points (-4,1) and (8,4) will be written.
1
Find the Slope
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Given two points on a line, the slope of the line can be found by using the Slope Formula. In this case, the coordinates (-4,1) and (8,4) can be substituted in place of (x1,y1) and (x2,y2), respectively.
m=0.25
The slope m of the line passing through the two points is 0.25.
2
Replace m With the Slope
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Now that the value of the slope is known, it can be substituted for m in the slope-intercept form of an equation.
3
Find b Using a Point
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Next, the y-intercept can be found by substituting either of the given points into the equation and solving for b. In the considered example, (8,4) can be used. Substitute its coordinates into the equation from Step 2 and solve for b.
y=0.25x+b
4=0.25(8)+b
4=2+b
2=b
b=2
Therefore, the y-intercept is 2.
4
Write the Equation
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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.
The equation is now complete.

Example

Write the equation of the line that passes through the given points

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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 expand_more
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
1=m3+4
1=3m+4
-3=3m
-1=m
m=-1
The slope of the new line is m=-1. Thus, we can write the complete equation as
y=-x+4.

Method

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

To write the equation in slope-intercept form of the graph of a line, the y-intercept b and the slope m of the line must be found.
The following method can be used. As an example, consider the line shown.
A line on a coordinate plane
1
Find the y-Intercept
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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).

The y-intercept of the line is identified on the graph
2
Replace b With the y-Intercept
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The y-coordinate of the y-intercept can be substituted into the slope-intercept form equation for b. In the considered example, substituting -4 for b results in the following equation.
3
Find the Slope
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Next, the slope of the line must be determined. By the definition, the slope of a line is expressed as the quotient of the rise and run of the line.
Here, rise is the vertical distance between two points, and run is the horizontal distance. To find the slope, use any two points. On the diagram, the previously-found y-intercept and the point (2,2) were chosen.
Rise and run between the two chosen points are determined
As seen above, the rise equals 6 and the run equals 2. By substituting these values into the formula, the slope can be calculated.
4
Replace m With the Slope
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The complete equation of the line can now be written by substituting the value of m into the equation from Step 2. In this case, substitute m=3.
The equation is now complete.
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