Point-Slope Form of a Line

The figure below shows the line that has slope and contains the point (3, 5).

You probably know by now that the slope is the same no matter which two points of the line are used to calculate it. So if we find the slope m for this line using an arbitrary point of the line, say (x, y), and the specific point (3, 5), we get

Because the slope of this line is , we can write

Multiplying each side by x - 3, we get

Because (x, y) was an arbitrary point on the line, this equation is satisfied by every point on the line.

If we use (x₁, y₁) as the specific point and (x, y) as an arbitrary point on a line with slope m, we can write

Multiplying each side of this equation by x - x₁ gives us the point-slope form of the equation of the line.

Point-Slope Form

The equation of the line through (x₁, y₁) with slope m in point-slope form is y - y₁ = m(x - x₁).

Example 1

Writing an equation for a line given a point and the slope

Find an equation for the line through (-2, 5) with slope -3 and solve it for y.

Solution

Use x₁ = -2, y₁ = 5, and m = -3 in the point-slope form:

y - 5 = -3[x - (-2)]

Now solve the equation for y:

If you know two points on a line, then you can graph the line (two points determine a line). In the next example we will see that two points of a line also determine an equation for the line.

Example 2

Writing an equation for a line given two points on the line

Find an equation for the line through (3, -2) and (-1, 1) and solve it for y.

Solution

We are not given the slope, but we can find it because the points (3, -2) and (-1, 1) are on the line:

Now use this slope and one of the points, say (3, -2), to write the equation in pointslope form:

Note that we would get the same equation if we had used slope and the other point (-1, 1). Try it.

For the next example, recall that if a line has slope m, then the slope of any line perpendicular to it is , provided that m ≠ 0.

Example 3

An equation of a line perpendicular to another line

Line l goes through (2, 0) and is perpendicular to the line through (5, -1) and (-1, 3). Find the equation of line l and then solve it for y.

Solution

First find the slope of the line through (5, -1) and (-1, 3):

Because line l is perpendicular to this line, line l has slope . Now use (2, 0) and the slope in the point-slope formula to get the equation of line l:

Distributive property