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