Question

How can you find the equation of a line that is parallel/perpendicular to a given line

Answer 1

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 --> $y = 5x + 2$. 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, $y=mx+b$ ---> $3 = 5(2) + b$. Then you solve for b.

$3 = 5(2) + b$

$3 = 10 + b$

$3 - 10 = 10 - 10 + b$

$-7 = b$     $b=-7$

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 $y = 5x - 7$.

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 --> $y=5x+2$. The slope is 5, to find the opposite reciprocal of the slope you use n --> $-\frac{1}{n}$. In this case, it would be $-\frac{1}{5}$. The opposite reciprocal of the slope of line a is $-\frac{1}{5}$, 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 $-\frac{1}{5}$ and the point is (1, 7), use the Slope-Intercept Formula, $y=mx+b$ ---> $7=-\frac{1}{5}(1) + b$. Then you solve for b.

$7=-\frac{1}{5}(1) + b$

$7=-\frac{1}{5} + b$

$7+\frac{1}{5}=-\frac{1}{5} + \frac{1}{5} + b$

$7\frac{1}{5} =b$     $b=7\frac{1}{5}$

The y-intercept of line b is $7\frac{1}{5}$.

3) Finally, plugin the slope and the y-intercept into the Slope-Intercept Form.

Example: The slope of line b is $-\frac{1}{5}$ and the y-intercept is $7\frac{1}{5}$. The equation of line b is $y=-\frac{1}{5}x+7\frac{1}{5}$.