mathleaks.com mathleaks.com Start chapters home Start History history History expand_more Community
Community expand_more
menu_open Close
{{ filterOption.label }}
{{ item.displayTitle }}
{{ item.subject.displayTitle }}
arrow_forward
No results
{{ searchError }}
search
Expand menu menu_open home
{{ courseTrack.displayTitle }}
{{ statistics.percent }}% Sign in to view progress
{{ printedBook.courseTrack.name }} {{ printedBook.name }}
search Use offline Tools apps
Login account_circle menu_open
close expand
Writing Linear Equations

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 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

Find the slope
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.
m=0.25
The slope m of the line passing through the two points is 0.25.

2

Replace m with the slope
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
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.
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
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.
fullscreen
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.

Show Solution
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
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 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,-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=-4 gives
y=mx4.

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 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

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=3x4.
arrow_left
arrow_right
{{ 'mldesktop-placeholder-grade-tab' | message }}
{{ 'mldesktop-placeholder-grade' | message }} {{ article.displayTitle }}!
{{ grade.displayTitle }}
{{ 'ml-tooltip-premium-exercise' | message }}
{{ 'ml-tooltip-programming-exercise' | message }} {{ 'course' | message }} {{ exercise.course }}
Test
{{ focusmode.exercise.exerciseName }}
{{ 'ml-btn-previous-exercise' | message }} arrow_back {{ 'ml-btn-next-exercise' | message }} arrow_forward
arrow_left arrow_right