Problem
Find the equation of a line through (1, 1) perpendicular to a line passing through (1, 2) and (3, 5).
Solution
Just to keep things straight, let's call the line whose equation we're looking for y1, and let's call the line through (1, 2) and (3, 5) y2. To write the equation for y1, we need two pieces of information: either a point and a slope, or two points on the line. We are given a point on y1, and y1 is described as being perpendicular to another line, so we will look for the slope of y1. Therefore, we must find the slope of y2.
Since we have two points on y2, the slope m of y2 is equal to
Since y1 is perpendicular to y2, the slope of y1 is the negative reciprocal of the slope of y2.
Now that we have the slope of y1, we know the equation for y1 is
where b is the y-intercept. We solve for b by plugging in the x- and y-values of the point that we know lies on y1:
Solve for b:
The equation of y1 is