Equation of a Line Through Point and Perpendicular to Another Line
Problem Find the equation of the line through the point (3, 7) which is perpendicular to the line y = 4x − 5.
Solution
The equation of the line they are looking for is of the form y = mx + b, where m is the slope and b is the y-intercept.
There are two things to find in order to get the equation: the slope and the y-intercept.
1. Slope
Easy. Slope is the inverse of the slope of the given equation, and of the opposite direction (add a negative sign to the slope).
So, the slope of the given equation is 4. The inverse means flip it. So, to flip it we have 1/4. Oppisite direction is adding negative sign so the slope of the desired equation is m = −1/4.
Why do we do this? Because the new equation is perpendicular to the old one. If it is perpendicular, this implies that we 1. take the inverse of the original equation's slope and 2. add a negative sign.
2. The y-intercept
The y-intercept is just plug and chug. Using the found slope, we have the equation y = (−1/4)x + b.
The equation passes through the point (3, 7), which is of the form (x, y), so we can plug these two values into our equation.
Doing so,
7 = (−1/4)*3 + b
7 = −3/4 + b
28/4 = −3/4 + b
31/4 = b
b = 31/4
We have found our y-intercept, and so we can form our answer.
The equation of the line through the point (3, 7) which is perpendicular to the line y = 4x − 5 is: