In the Slope-Intercept Form, m is the slope and b is the y-intercept.
Finding The Equation of a Line Parallel to the Given Line: Let the given line be line a and the parallel line be line b.
1) Find the slope of line a.
Example: Line a --> [tex]y = 5x + 2[/tex]. The slope is 5.
2) Find the y-intercept of line b.
To do this, you need to be given the coordinates of a point located on line b.
Example: The slope is 5 and the point is (2, 3), use the Slope-Intercept Formula, [tex]y=mx+b[/tex] ---> [tex]3 = 5(2) + b[/tex]. Then you solve for b.
[tex]3 = 5(2) + b[/tex]
[tex]3 = 10 + b[/tex]
[tex]3 - 10 = 10 - 10 + b[/tex]
[tex]-7 = b[/tex] [tex]b=-7[/tex]
The y-intercept of line b is -7
3) Finally, plugin the slope and the y-intercept into the Slope-Intercept Form.
Example: The slope of line b is 5 and the y-intercept is -7. The equation of line b is [tex]y = 5x - 7[/tex].
Finding The Equation of a Line Perpendicular to The Given Line: Let the given line be line a and the perpendicular line be line b.
1) Find the opposite reciprocal of the slope of line a.
Example: Line a --> [tex]y=5x+2[/tex]. The slope is 5, to find the opposite reciprocal of the slope you use n --> [tex]-\frac{1}{n}[/tex]. In this case, it would be [tex]-\frac{1}{5}[/tex]. The opposite reciprocal of the slope of line a is [tex]-\frac{1}{5}[/tex], which would be the slope of line b.
2) Find the y-intercept of line b.
To do this, you need to be given the coordinates of the point where line a and line b both intersect.
Example: The slope of line b is [tex]-\frac{1}{5}[/tex] and the point is (1, 7), use the Slope-Intercept Formula, [tex]y=mx+b[/tex] ---> [tex]7=-\frac{1}{5}(1) + b[/tex]. Then you solve for b.
[tex]7=-\frac{1}{5}(1) + b[/tex]
[tex]7=-\frac{1}{5} + b[/tex]
[tex]7+\frac{1}{5}=-\frac{1}{5} + \frac{1}{5} + b[/tex]
[tex]7\frac{1}{5} =b[/tex] [tex]b=7\frac{1}{5}[/tex]
The y-intercept of line b is [tex]7\frac{1}{5}[/tex].
3) Finally, plugin the slope and the y-intercept into the Slope-Intercept Form.
Example: The slope of line b is [tex]-\frac{1}{5}[/tex] and the y-intercept is [tex]7\frac{1}{5}[/tex]. The equation of line b is [tex]y=-\frac{1}{5}x+7\frac{1}{5}[/tex].
