Finding Equations of Lines

There are two formulas that can be used to find the equation of a line. One is the slope-intercept formula which can be solved for m and b. The other is the point-slope formula. Since this seems to be the method used in most modern textbooks, it is the one that will be demonstrated here.

Key Idea

A key point to remember when asked to find the equation of a line is that you will always need the slope and a point on the line. No matter what information is given, there will be some way to determine the slope and a point on the line. Keeping that in mind helps you focus on the key concepts for finding equations of lines no matter how different the problems may look. Always find the slope and a point and plug them into the point-slope formula to get the equation of the line.

Given Slope and Point

Example

Find the equation of the line that passes through the point (-4, 9) with slope -5.

Solution

In the point-slope formula let x1 = -4, y1 = 9, and m = -5 .

y - 9 = -5(x - -4)

y - 9 = -5(x + 4)

Distribute the -5 to remove the parentheses. y - 9 = -5x - 20

Then add 9 to both sides to solve for y. y = -5x - 11

Given Two Points

When given two points, find the slope and then proceed as in part A above.

Example

Find an equation of the line that passes through the points (-1, 4) and (7, 5).

Solution

First find the slope, m.

Pick either point for (x1, y1) and use the point-slope formula.

Using the point (7, 5) gives

The equation could also be written as x - 8y = -33 or in standard form as x - 8y + 33 = 0.

You should be able to recognized various forms of an equation as being equivalent.

Check

To help find errors substitute the point you did not use, in this case (-1,4), into the equation and see if it works.

Given a Point and Parallel or Perpendicular Line

Often problems ask for the equation of a line through a given point that is parallel or perpendicular to a given line. Notice we have the point needed for the point-slope formula. The slope can be found by using the appropriate fact.

  1. Parallel lines have the same slope.
  2. Perpendicular lines have slopes that are negative reciprocals.

Example

Find an equation of the line that passes through the point (3, -2) and is parallel to the line with equation

y = 4x - 7.

Solution

The slope of the line can be read directly from the equation. m = 4

Substituting into the point-slope formula gives y - -2 = 4(x - 3)

Solve for y to get:

y + 2 = 4x - 12

y = 4x - 14

Example

Find an equation of the line that passes through the point (3, -2) and is perpendicular to the line with equation y = 4x - 7.

Solution

The slope of the given line is 4 so the slope of the line for which we are finding the equation is -¼.

Substitution into the point-slope formula gives y - -2 = -¼(x - 3)

Solve for y to get: