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.

**Example**

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

**Solution**

In the point-slope formula let x_{1} = -4, y_{1} = 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**

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 (x_{1}, y_{1}) 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.

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.

- Parallel lines have the same slope.
- 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: