- The General Form of a Line
- The Slope-Intercept Form of a Line
- Slope and Intercept
- The Formula for the Slope of a Line

Inside Collection (Textbook): Basic Mathematics Review

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter the student is shown how graphs provide information that is not always evident from the equation alone. The chapter begins by establishing the relationship between the variables in an equation, the number of coordinate axes necessary to construct its graph, and the spatial dimension of both the coordinate system and the graph. Interpretation of graphs is also emphasized throughout the chapter, beginning with the plotting of points. The slope formula is fully developed, progressing from verbal phrases to mathematical expressions. The expressions are then formed into an equation by explicitly stating that a ratio is a comparison of two quantities of the same type (e.g., distance, weight, or money). This approach benefits students who take future courses that use graphs to display information. The student is shown how to graph lines using the intercept method, the table method, and the slope-intercept method, as well as how to distinguish, by inspection, oblique and horizontal/vertical lines. Objectives of this module: be more familiar with the general form of a line, be able to recognize the slope-intercept form of a line, be able to interpret the slope and intercept of a line, be able to use the slope formula to find the slope of a line.

- The General Form of a Line
- The Slope-Intercept Form of a Line
- Slope and Intercept
- The Formula for the Slope of a Line

We have seen that the general form of a linear equation in two variables is

This equation is of the form *Note:* The fact that we let

The following examples illustrate this procedure.

Solve

This equation is of the form

Solve

This equation is of the form

Solve

This equation is of the form

A linear equation in two variables written in the form

The following equations *are* in slope-intercept form:

The following equations *are not* in slope-intercept form:

The following equation are in slope-intercept form. In each case, specify the slope and

When the equation of a line is written in slope-intercept form, two important properties of the line can be seen: the *slope* and the *intercept*. Let's look at these two properties by graphing several lines and observing them carefully.

Graph the line

0 | ||

4 | 1 | |

Looking carefully at this line, answer the following two questions.

At what number does this line cross the

The line crosses the

Place your pencil at any point on the line. Move your pencil exactly *one* unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation?

After moving horizontally one unit to the right, we must move exactly one vertical unit up. This number is the coefficient of

Graph the line

0 | 1 | |

3 | 3 | |

Looking carefully at this line, answer the following two questions.

At what number does this line cross the

The line crosses the

*one* unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation?

After moving horizontally one unit to the right, we must move exactly

Graph the line

0 | ||

3 | ||

2 |

Looking carefully at this line, answer the following two questions.

At what number does the line cross the

The line crosses the

*one* unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation?

In the graphs constructed in Sample Set B and Practice Set B, each equation had the form

At what number does the line cross the

In each case, the line crosses the

*one* unit horizontally to the right. Now, how many units straight up or down must you move your pencil to get back on the line? Do you see this number in the equation?

To get back on the line, we must move our pencil exactly

The number

Since the equation *slope-intercept* form.

The slope-intercept form of a straight line is

The slope of the line is

The word *slope* is really quite appropriate. It gives us a measure of the steepness of the line. Consider two lines, one with slope *one* unit to the right. To get back to one line we need only move vertically

Find the slope and the

The line is in the slope-intercept form

The line is in slope-intercept form

The equation is written in general form. We can put the equation in slope-intercept form by solving for

Now the equation is in slope-intercept form.

Find the slope and

Solving for

We have observed that the slope is a measure of the steepness of a line. We wish to develop a formula for measuring this steepness.

It seems reasonable to develop a slope formula that produces the following results:

Steepness of line

Consider a line on which we select any two points. We’ll denote these points with the ordered pairs

The difference in

We are comparing changes. We see that we are comparing

This is a comparison and is therefore a ratio. Ratios can be expressed as fractions. Thus, a measure of the steepness of a line can be expressed as a ratio.

The slope of a line is defined as the ratio

Mathematically, we can write these changes as

The slope of a nonvertical line passing through the points

For the two given points, find the slope of the line that passes through them.

Looking left to right on the line we can choose

This line has slope 2. It appears fairly steep. When the slope is written in fraction form,

Notice that as we look left to right, the line rises.

Looking left to right on the line we can choose

This line has slope

Notice that in examples 1 and 2, both lines have positive slopes, *rise* as we look left to right.

Looking left to right on the line we can choose

This line has slope

When the slope is written in fraction form,

Notice also that this line has a negative slope and declines as we look left to right.

This line has 0 slope. This means it has *no* rise and, therefore, is a horizontal line. This does not mean that the line has no slope, however.

This problem shows why the slope formula is valid only for nonvertical lines.

Since division by 0 is undefined, we say that vertical lines have undefined slope. Since there is no real number to represent the slope of this line, we sometimes say that vertical lines have *undefined slope*, or *no slope*.

Find the slope of the line passing through

Find the slope of the line passing through

The line has slope

Compare the lines of the following problems. Do the lines appear to cross? What is it called when lines do not meet (parallel or intersecting)? Compare their slopes. Make a statement about the condition of these lines and their slopes.

The lines appear to be parallel. Parallel lines have the same slope, and lines that have the same slope are parallel.

Before trying some problems, let’s summarize what we have observed.

The equation

The slope,

The formula for finding the slope of a line through any two given points

The fraction

As we look at a graph from left to right, lines with positive slope rise and lines with negative slope decline.

Parallel lines have the same slope.

Horizontal lines have 0 slope.

Vertical lines have undefined slope (or no slope).

For the following problems, determine the slope and

For the following problems, find the slope of the line through the pairs of points.

Do lines with a positive slope rise or decline as we look left to right?

Do lines with a negative slope rise or decline as we look left to right?

decline

Make a statement about the slopes of parallel lines.

For the following problems, determine the slope and

For the following problems, find the slope of the line through the pairs of points. Round to two decimal places.

*((Reference))* Simplify

*((Reference))* Solve the equation

*((Reference))* When four times a number is divided by five, and that result is decreased by eight, the result is zero. What is the original number?

10

*((Reference))* Solve

*((Reference))* Graph the linear equation

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Comments:"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"