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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.

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

To write linear equations in slope-intercept form, y=mx+b, y=mx+b, the slope, m,m, and the yy-intercept, b,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)(\text{-} 4,1) and (8,4).(8,4).


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 (8,4)(8,4) and (-4,1)(\text{-} 4,1) can be substituted in place of (x1,y1)(x_1,y_1) and (x2,y2),(x_2,y_2), respectively.
m=y2y1x2x1m = \dfrac{y_2-y_1}{x_2-x_1}
m=14-48m = \dfrac{{\color{#0000FF}{1}}-{\color{#009600}{4}}}{{\color{#0000FF}{\text{-} 4}}-{\color{#009600}{8}}}
m=-3-12m=\dfrac{\text{-} 3}{\text{-} 12}
The slope of the line passing through the two points is m=0.25.m=0.25.


Replace mm with the slope

The equation y=mx+by=mx+b can be re-written with m=0.25.m=0.25. This gives y=0.25x+b. y=0.25x+b.


Find bb using a point
Next, the yy-intercept can be found by replacing xx and yy in the equation with either of the given points. Then, solving for bb gives the yy-intercept. Here, the arbitrarily chosen point that will be used is (8,4).(8,4). Therefore, substitute x=8x=8 and y=4y=4 into the equation from Step 2.
4=0.258+b{\color{#009600}{4}}=0.25\cdot {\color{#0000FF}{8}}+b
Thus, the yy-intercept is b=2.b=2.


Write the equation

Lastly, the complete equation in slope-intercept form can be written by replacing the value of the yy-intercept found above. Here, b=2b=2 will be substituted into the equation from Step 2. This gives y=0.25x+2. y=0.25x+2.


Write the equation of the line that passes through the point (3,1)(3,1) and has the same yy-intercept as the line y=9x+4.y=9x+4.

Show Solution
A line in slope-intercept form is given by the equation y=mx+b, y=mx+b, where mm is the slope of the line and bb is the yy-intercept. The line y=9x+4y=9x+4 has a yy-intercept of (0,4).(0,4). We want our line to have the same yy-intercept. Therefore, the equation of the new line must also have the value b=4.b=4. This gives y=mx+4. y=mx+4. Our line must also pass through the point (3,1).(3,1). We can solve for mm in the equation above by substituting this point for xx and y.y.
1=m3+4{\color{#009600}{1}}=m\cdot {\color{#0000FF}{3}} + 4
1=3m+41= 3m + 4
-3=3m\text{-} 3 = 3m
-1=m\text{-} 1 = m
m=-1m = \text{-} 1
The slope of the new line is m=-1.m=\text{-} 1. Thus, we can write the complete equation as y=-x+4. y=\text{-} x+4.

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, y=mx+b,

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


Find the yy-intercept

The yy-intercept is the point where the graph intersects the yy-axis. From the diagram, it can be seen that the yy-intercept is (0,-4).(0,\text{-} 4).


Replace bb with the yy-intercept

The yy-coordinate of the yy-intercept can be substituted into y=mx+by=mx+b for b.b. Here, substituting b=-4b=\text{-} 4 gives y=mx4. y=mx-4.


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=riserun, m=\dfrac{\text{rise}}{\text{run}}, 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 yy-intercept and the arbitratily chosen (2,2).(2,2).

From the lines drawn, it can be seen that the rise=6\text{rise}=6 and the run=2.\text{run}=2. Therefore, the slope is m=62=3. m=\dfrac{6}{2}=3.


Replace mm with the slope

The complete equation of the line can now be written by substituting the value of mm into the equation from Step 2. Here, substitute m=3.m=3. y=3x4. y=3x-4.

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