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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, the slope, and the -intercept, 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 and

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 and can be substituted in place of and respectively.
The slope of the line passing through the two points is

2

Replace with the slope

The equation can be re-written with This gives

3

Find using a point
Next, the -intercept can be found by replacing and in the equation with either of the given points. Then, solving for gives the -intercept. Here, the arbitrarily chosen point that will be used is Therefore, substitute and into the equation from Step 2.
Thus, the -intercept is

4

Write the equation

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

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Exercise

Write the equation of the line that passes through the point and has the same -intercept as the line

Show Solution
Solution
A line in slope-intercept form is given by the equation where is the slope of the line and is the -intercept. The line has a -intercept of We want our line to have the same -intercept. Therefore, the equation of the new line must also have the value This gives Our line must also pass through the point We can solve for in the equation above by substituting this point for and
The slope of the new line is Thus, we can write the complete equation as
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,

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

1

Find the -intercept

The -intercept is the point where the graph intersects the -axis. From the diagram, it can be seen that the -intercept is

2

Replace with the -intercept

The -coordinate of the -intercept can be substituted into for Here, substituting gives

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 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 -intercept and the arbitratily chosen

From the lines drawn, it can be seen that the and the Therefore, the slope is

4

Replace with the slope

The complete equation of the line can now be written by substituting the value of into the equation from Step 2. Here, substitute

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