A. Linear equations in one Variable

A linear equation in one variable is an equation that involves only the first power of the variable (typically represented by ) and no division by any expression involving the variable. Here are some examples of linear and nonlinear equations.

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A linear equation can be solved by performing the following steps:

Not every equation will require all four steps.

Example 1. Solve for .

Example 2. Solve for .

Practice Exercises

Solve the following equations for .

B. Plotting Points in the Plane

To form a coordinate plane, place two number lines at right angles, so they cross at 0. The number lines are called axes. Usually, the horizontal axis is labeled as the -axis and the vertical axis is labeled as the -axis. A point in the coordinate plane is specified by an ordered pair (, ). To plot an ordered pair, such as (3, 2), draw a vertical line through the -axis at and a horizontal line through the -axisat . The point of intersection of these two lines is the point (3, 2).

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Example 1. Plot the points (2, 3), (, 4), (4, ), and ().

Practice Exercises

C. Distance and Midpoint between Two Points in the Plane

Given two points in the plane, sometimes we will want to calculate the distance between them; other times, we will want to place a point midway between the two points. Given two points in the plane, () and (), we use the following formula to find the distance between them.

Distance between :

Example 1. Find the distance between the points (1, 2) and (6, 5).

To find the midpoint between two numbers on a number line, simply average the two numbers.

Midpoint between and : Midpoint

Example 2. Determine the midpoint between 2 and 12.

The distance between 2 and 12 is 10. So the midpoint should be , or 5, units from both 2 and 12. Using the midpoint formula we get:

Notice that 7 is 5 units above 2 and 5 units below 12.

We can also find the midpoint of a line segment between two points () and (). The - and -coordinates of the midpoint will be the average of the - and -coordinates of the two points, respectively.

Midpoint between () and () is

Example 3. Find the midpoint between the points (1, 2) and (6, 5). Then find the distance between (1, 2) and the midpoint and the distance between the midpoint and (6, 5). What is true of these two distances?

The distance between (1, 2) and is

The distance between and (6, 5) is

These two distances are equal, half the distance between (1, 2) and (6, 5).

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Practice Exercises

D. Linear equations in TWO Variables

A linear equation in two variables (typically and ) can be written in the form . A solution to such an equation is an ordered pair, (, ), that satisfies the equation.

Example 1. Verify that (4, ) is a solution to .

Substitute 4 for and for , and then simplify to show that the expression on the left side of the equation is equal to 6, the number on the right side.

Example 2. Show that (1, 1) is not a solution to .

A line is a visual representation of the linear equation’s solution set and can be graphed by drawing a line through two distinct points in the solution set. When the linear equation is in the form , an efficient method of graphing is to graph by intercepts. The -intercept can be found by substituting and then solving for . The -intercept can be found by substituting and then solving for . To graph , draw a line through the - and -intercepts.

Example 3. Determine the and intercepts for . Draw a line that represents the solution set.

To find the -intercept, substitute and solve for :

The -intercept is the ordered pair (0, 3).

To find the -intercept, substitute and solve for :

The -intercept is the ordered pair (2, 0).

To graph , plot the two intercepts and then draw a line through them as shown below. Notice that (4, ) (see Example 1) is on the line, but (1, 1) (see Example 2) is not.

Two special kinds of lines of interest are horizontal and vertical lines. A horizontal line can be written in the form ( is where the line crosses the -axis). A vertical line can be written in the form ( is where the line crosses the -axis).

Example 4. Graph the solution set of .

Start by simplifying the equation and getting all variables to one side of the equation.

The graph of the solution set is a horizontal line with -inter- cept at (0, 2).

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Example 5. Graph the line .

The graph is a vertical line with x-intercept (1, 0)

Practice Exercises

E. slope of a Line

The slope of a line is defined to be the ratio of the vertical change to the horizontal change. If we think of slope as , the slope of the line below would be .

Given two points () and (), the formula to calculate slope is .

Example 1. Calculate the slope of a line that passes through the points (1, 3) and (4, 0).

By letting and , the slope of the line below would be

Given a graph of a line, we can determine visually whether the slope is positive or negative. If the -values increase as the -values increase, as is the case in the graph that appears before Example 1, then the slope is positive. On the other hand, if the -values decrease as the -values increase, as is the case for the graph in Example 1, then the slope is negative. Horizontal lines have a slope of 0. Vertical lines have undefined slope.

Example 2. Graph the line that passes through the points (1, 3) and (4, 3). Determine its slope.

Example 3. Graph the line that passes through the points (1, 3) and (1, 1). Determine its slope.

(For additional help in computing slopes, see Example 2 in Algebra Review III, item A, Using Formulas, page AR-7.)

Practice Exercises

For Practice Exercises 1–4, determine the slope of the line that passes through the given points.

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F. Graphing a Line in slope-Intercept Form

When a linear equation is written in the form , it is said to be in slope-intercept form. In this form, is the location where its graph intersects the vertical axis, which is generally referred to as the -intercept. This intercept represents one point on the graph of the line. In order to graph a line, two points are required. A second point can be obtained by using the slope.

Example 1. Graph the line by first plotting the -intercept and then using the slope to find a second point on the line.

In the graph below, first plot the -intercept, (). Using the slope of , find a second point on the line by starting at () moving up 2 units and to the right 3 units, which gives the point (3, 1). Draw a line through () and (3, 1).

Example 2. Graph the line .

First, plot the -intercept, (0, 2). Instead of using the slope to obtain a second point, it may be easier to evaluate the formula with a value for other than . For example, if we substitute , we find that . This corresponds to the point (4, 0). By drawing a line through the -intercept and the point (4, 0), we have the following graph:

Calculator Note: You can use a graphing calculator to graph a line in the form . Here’s how:

Example 3. Using a graphing calculator, graph in the standard viewing window.

Following the note above, enter opposite Y1. Your graph should be similar to the screenshot shown below.

Practice Exercises

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For Practice Exercises 7 and 8, use a graphing calculator to graph the equations.

G. Linear Inequalities in TWO Variables

To graph linear inequalities, we first graph the line, determine whether the line should be drawn solid or dashed, and then determine which side of the line to shade.

Example 1. Graph in the first quadrant.

This line has a -intercept of (0, 6) and an -intercept of (4,0). We draw a solid line because of the inclusive inequality symbol , which indicates that the points on the line are part of the solution. Next, we test to see if (0, 0) is in the solution set: , or , is a true statement. Therefore, we shade the region on the side of the line containing our test point.

Calculator Note: To graph a linear inequality, you will first need to solve the inequality for . In other words, express the inequality in the form , where the inequality sign could be any of the four possibilities: .

The following algebraic manipulations do not change the solution set of an inequality:

Example 2. Use a graphing calculator to graph .

Unlike in Example 1, for Example 2 we did not restrict the solution set to the first quadrant.

Note: You will need to determine whether to draw a solid line or a dotted line. In this case, a solid line should be drawn because of the inclusive inequality .

Example 3. Use algebraic manipulation to solve the following inequality for . Use a graphing calculator to graph the solution set.

Practice Exercises

H. systems of Linear Equations and Inequalities

Systems of Linear Equations

Two or more linear equations that are to hold true at the same time are called a system of linear equations. Systems of linear equations can be represented by graphing the linear equations and determining the point of intersection, if one exists. The point of intersection is the solution to the system of linear equations.

Here is a general strategy for solving a system of two equations in two variables:

Example 1. Solve the system:

The solution is (3, 2).

Finally, we check that our solution is correct:

Example 2. Represent graphically the system of equations from Example 1.

We can use the graph-by-intercept approach (see Algebra Review IV, item D, Linear Equations in Two Variables, page AR-13) to graph the two linear equations. Below are graphs of equations (1) and (2), along with their corresponding x- and y-intercepts. The solution to the system of equations is the point of intersection of the two lines, which appears to be (3, 2).

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Calculator Note: Next, we find the solution to the system of linear equations in Example 1 using a TI-84 graphing calculator.

First we solve equations (1) and (2) for , which results in the following equations:

Next, use a TI-84 graphing calculator to graph equations and :

Now you need to find the point of intersection. Here’s how:

Systems of Linear Inequalities

If you are given more than one inequality, then you have a system of linear inequalities. In graphing a system of linear inequalities, you look for the region that is common to the graphs of all the individual inequalities.

Example 3. Graph the solution set of .

Begin by graphing the equations . We have drawn solid lines in the graph that follows because each of the inequalities is inclusive ( or ). We can see that the shaded region satisfies all three inequalities. In addition, the three points of intersection are shown. These were found by examining the three lines two at a time.

Calculator Note: Since is not a function (it fails the vertical line test), you cannot graph it using the menu on a TI-84 graphing calculator. Graph the other functions first and approximate the graph of by pressing (for DRAW) and then (for Vertical), followed by . Use the left or right arrow keys to move the vertical line as close as possible to its desired location. (You won’t be able to get it exact.)

Practice Exercises

For Practice Exercises 2 and 3, use algebraic procedures to solve the system of equations.

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