## Easy Algebra Step-by-Step: Master High-Frequency Concepts and Skills for Algebra Proficiency—FAST! (2012)

### Chapter 18. The Equation of a Line

In this chapter, you determine the equation of a line. The basic graph of all of mathematics is the straight line. It is the simplest to draw, and it has the unique property that it is completely determined by just two distinct points. Because of this unique property, it is a simple matter to write the equation of a line given just two items of critical information.

There are three common methods for determining the equation of a line.

**Determining the Equation of a Line Given the Slope and y-Intercept**

This is the simplest of the methods for determining the equation of a line. You merely use the slope-*y*-Intercept form of the equation of a line:

**Problem** Given the slope *m* = 3 and the *y*-Intercept *y* = 5, write the equation of the line.

**Solution**

*Step 1*. Recalling that the slope-*y*-Intercept form of the equation of a line is *y* = *mx* + *b*, write the equation.

The equation of the line is (You can see why this is the simplest method!)

**Problem** Given the slope and the *y*-Intercept *y* = –2, write the equation of the line.

**Solution**

*Step 1*. Recalling that the slope-y-intercept form of the equation of a line is write the equation.

The equation of the line is

**Determining the Equation of a Line Given the Slope and One Point on the Line**

For this method, you use the point-slope equation , where (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) are points on the line.

Watch your signs when you use the point-slope equation.

**Problem** Given the slope *m* = 2 and a point (3, 2) on the line, write the equation of the line.

**Solution**

*Step 1*. Let (*x*, *y*) be a point on the line different from (3, 2), then substitute the given information into the point-slope formula: .

*Step 2*. Solve the equation for *y* to get the slope-*y*-Intercept form of the equation.

*y* = 2*x* – 4 is the equation of the line.

**Problem** Given the slope and a point (–1, 3) on the line, write the equation of the line.

**Solution**

*Step 1*. Let (*x*, *y*) be a point on the line different from (–1, 3), then substitute the given information into the point-slope formula: .

*Step 2*. Solve the equation for *y* to get the slope-*y*-Intercept form of the equation.

**Problem** Given the slope *m* = –2 and a point (0, 0) on the line, write the equation of the line.

**Solution**

*Step 1*. Let (*x*, *y*) be a point on the line different from (0, 0), then substitute the given information into the point-slope formula:

*Step 2*. Solve the equation for *y* to get the slope-*y*-Intercept form of the equation.

*y* = –2*x* is the equation of the line.

**Determining the Equation of a Line Given Two Distinct Points on the Line**

You also use the point-slope equation with this method.

**Problem** Given the points (3, 4) and (1, 2) on the line, write the equation of the line.

**Solution**

*Step 1*. Use the two points to determine the slope using the point-slope equation.

*Step 2*. Now use the point-slope formula and one of the given points to finish writing the equation. Let (*x*, *y*) be a point on the line different from, say, (3, 4).

*Step 3*. Solve the equation for *y* to get the slope-*y*-Intercept form of the equation.

**Problem** Given the points (–1, 4) and (3, –7) on the line, write the equation of the line.

**Solution**

*Step 1*. Use the two points to determine the slope using the point-slope equation.

*Step 2*. Now use the point-slope formula and one of the given points to finish writing the equation. Let (*x*, *y*) be a point on the line different from, say, (3, –7).

*Step 3*. Solve the equation for *y* to get the slope-*y*-Intercept form of the equation.

When two points are known, it does not make any difference which one is chosen to finish writing the equation.

**Exercise 18**

__1__. Given the slope *m* = 4 and the *y*-Intercept *y* = 3, write the equation of the line.

__2__. Given the slope *m* = –3 and the *y*-Intercept *y* = –3, write the equation of the line.

__3__. Given the slope and the *y*-Intercept *y* = 0, write the equation of the line.

__4__. Given the slope *m* = 2 and a point (1, 1) on the line, write the equation of the line.

__5__. Given the slope *m* = –1 and a point (2, 3) on the line, write the equation of the line.

__6__. Given the slope and a point (0, 1) on the line, write the equation of the line.

__7__. Given the points (2, 4) and (1, 2) on the line, write the equation of the line.

__8__. Given the points (–1, 2) and (1, 2) on the line, write the equation of the line.

__9__. Given the points (2, –1) and (1, 0) on the line, write the equation of the line.

__10__. Given the points (4, 4) and (6, 6) on the line, write the equation of the line.