### Unit 5: Graphs of Linear Equations and Inequalities

This unit is an introduction to graphing relationships between the two quantities on the coordinate plane. A graph helps visualize how one quantity depends on another. In this course, you will only graph relationships that can be described by a linear equation, and their graphs are always straight lines. Graphing is an important tool that aids in analyzing relationships, both in abstract and in applied math. The material in this unit will help you become comfortable with graphing pairs of numbers on the coordinate plane and understand how the equations and relationships can be represented by lines. As an example, you might want to graph how the location of a train depends on the time since the train departed from the station. If the train is moving with constant speed, this graph would be a straight line. The slant of this line (which, as you will learn in this unit, is called the slope) will depend on the speed: the greater the speed, the steeper the line. If the line is going up (looking from left to right), it tells you that the distance is growing with time, which means that the train is moving away from the station. Otherwise, if the line is going down, it tells you that the distance is decreasing, which means that the train is approaching the station. You will see that a lot of information about the train's journey can be gathered from just one graph.

**Completing this unit should take you approximately 24 hours.**

Upon successful completion of this unit, you will be able to:

- graph points with given coordinates on the rectangular coordinate plane;
- determine coordinates of a point on the rectangular coordinate system;
- determine whether a given ordered pair is a solution of the equation with two variables;
- find and graph solutions of the equation in two variable;
- graph a straight line given either its equation, or a slope and y-intercept;
- find slope and intercepts of a straight line given its equation or its graph;
- write the equation of a line passing through two given points;
- write the equation of a line with a given slope passing through a given point;
- locate on a coordinate plane all solutions of a given inequality in two variables;
- represent relationships between quantities as an equation or inequality in two variables; and
- interpret the meaning of slope and intercepts of the graph of a relationship between quantities.

### 5.1: Graphing Points in the Rectangular Coordinate Plane

Read this article, watch the videos embedded in the text, and try to play the interactive "Coordinate Plane Game," linked under the "Try This" heading. Then, complete practice problems 1-10.

Complete the exercise set. Drag the orange dot to the given point on the coordinate plane, then select the quadrant that contains this point from the choices on the right side of the page.

### 5.2: Linear Equations in Two Variables

Recall that in Unit 2 a

*solution*of an equation was defined as a number that make this equation a true statement. A solution of linear equations in 2 variables can be defined the same way, except now it is a pair of numbers (one for each variable) that makes the equation a true statement. In this subunit, you will learn how to find these solutions and represent them on a graph.

### 5.2.1: Ordered Pairs as Solutions of an Equation in Two Variables

Watch this video and take notes. This video is an introduction to a branch of mathematics known as Coordinate Geometry, which studies the connection between algebraic equations and properties of lines and curves representing them on the coordinate plane.

Read this page. These are the two main concepts you need to understand after reading this article:

- how to identify whether a pair of numbers is a solution of a given equation or inequality in two variables; and
- if you graph all solutions (ordered pairs of numbers) of a linear equation in two variables on a coordinate plane, they will create a straight line.

Practice applying these concepts by clicking on the "new problem" button at the end of the article, trying a problem, and checking your answer. Attempt to solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled "Click Here for a Randomly Generated Worksheet and Answers." The answers will be provided at the end of the worksheet.

- how to identify whether a pair of numbers is a solution of a given equation or inequality in two variables; and
Watch this video and take notes. In this video, you will see an example of how to determine whether a pair of numbers is a solution of a given equation in two variables.

Complete the exercise set. Determine which one of the given ordered pairs is a solution to the given (algebraically or graphically) linear equation.

### 5.2.2: Graphing Equations in Two Variables of the Form y = mx + b

Watch this video, take notes, and graph the lines shown on graph paper. This video shows how to create a graph of a linear equation by generating a few solutions (ordered pairs) and plotting them on a coordinate plane. All linear equations in this video are such that

*y*is expressed in terms of*x*:*y = mx + b*.Watch this video, take notes, and graph the line shown on graph paper. This is also an example of graphing a linear equation written in the form

*y = mx + b*, but the focus is on the fact that the coefficient in front of*x*is a fraction. Because we can pick any values of x to generate the ordered pairs for plotting, it is convenient to choose the ones that will produce integer values of*y*, which are easier to graph.

### 5.2.3: Graphing Equations in Two Variables of the Form Ax + By = C

Watch this video, take notes, and graph the line shown on graph paper. In this video, Sal Khan again graphs a linear equation by generating its solutions. However, this time an extra step is required: the original equation needs to be rewritten with the variable y on one side and everything else on the other side, so values of x can be plugged in and the values of y can be easily calculated. In addition to graphing a linear equation, the second video highlights the connection between the solutions of an equation and all points on the straight line, or the graph of the equation.

Watch this video, take notes, and graph the line shown on graph paper. This is another example of graphing a linear equation written in the form

*Ax + By = C*. Note how Dr. Sousa chooses the values of*x*to create a table of ordered pairs convenient for plotting.

### 5.2.4: Graphs of the Equations in the Forms x = C and y = C

### 5.2.4.1: Graph of a Vertical Line

Watch this video, take notes, and graph the line shown on graph paper.

### 5.2.4.2: Graph of a Horizontal Line

Watch this video, take notes, and graph the line shown on graph paper.

### 5.3: Slopes and Intercepts of a Straight Line

### 5.3.1: Intercepts of a Straight Line

Read this page. After reading and working through the examples, complete practice problems 1-10. Once you have completed the practice problems, check your answers against the answer key.

Read this page. This reading explains another method for identifying intercepts of the line from its equation. It also provides some examples of real-world applications of intercepts. After reading and working through the examples, complete practice problems 3-22. Use the video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. Find the value of the x- and y-intercepts using one of the methods you learned in the previous assignments, and enter it into the answer tab on the right side of the page.

### 5.3.2: Slope of a Straight Line

Slope is a major characteristic of a straight line. Slope is a number that indicates how steep the line is (relative to the horizontal) and whether the line is going up or down.

### 5.3.2.1: Definition of Slope and Slope Formula

Read this page. After reading and working through the examples, complete practice problems 4-23. Use the video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. You will be required to find the slope using the slope formula, or to identify a line with a given slope. Enter your answer into the tab on the right side of the page, or select one of the given choices.

### 5.3.2.2: Slopes of Parallel and Perpendicular Lines

Read this page. This reading defines parallel and perpendicular lines and illustrates the relationship between their slopes.

In the following exercises, you will apply the concepts and formulas from Dr. Burns' article. Work through Examples A, B, and C, and watch the video embedded in the text. Then, complete practice problems 1-4 and 11. Once you have completed the practice problems, check your answers against the answer key.

### 5.3.3: Applications: Real-World Interpretations of Slope and Intercepts

Slope represents the rate with which one quantity changes with respect to another quantity and the intercepts usually represent the starting and ending points of a process.

Watch these videos and take notes. The first video introduces the meaning of the slope as a rate of change. In the second video, the slope formula is used to calculate rate (in this case, rate of population growth per year). In the third video, the slope formula is again used to calculate rate (in this case, production cost per item).

### 5.4: Equations of Straight Line

You have probably noticed that linear equations are usually written in one of these forms:

*y = mx +b*, known as the slope-intercept form. As you will learn in this subunit,*m*in this equation is the slope and*b*is the*y*-intercept.*Ax + By = C*, known as the standard form. There is also another possible form, used to write an equation of the line with a given slope m passing through a given point (*x*). It is known as the point-slope form:_{1}, y_{1}*y - y*. In this subunit, you will be identifying the line's properties from these equations and converting equations from one form to another._{1}= m(x - x_{1})

### 5.4.1: Slope-Intercept Form

### 5.4.1.1: Graphing Equation of a Line in a Slope-Intercept Form

Read this page. Then, graph the linear equations given in practice problems 1-5 and 7, and complete practice problems 16-21. Use the video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against the answer key.

### 5.4.1.2: Writing Equation of a Line in a Slope-Intercept Form

Read this page. This article explains how to find an equation of a line in slope-intercept point when the slope and one point on the line are given, or when two points on the line are given. Practice writing equations by clicking on the "new problem" button at the end of the article, trying a problem, and checking your answer. Attempt to solve 10 problems. You can also create a worksheet of 10 problems by clicking on the button at the bottom of the page titled "Click Here for a Randomly Generated Worksheet and Answers." The answers will be provided at the end of the worksheet.

### 5.4.2: Point-Slope Form

Read this page. Then, complete practice problems 3-17. Use the video embedded in the text for guidance, if you need help. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise.

### 5.4.3: Standard Form

Read this page. Then, complete practice problems 4-27. If you need help, use the "Linear Equations in Standard Form" video embedded in the text for guidance.

Complete this exercise. You will have to convert the equations in standard form into equations in slope-intercept form and vice versa. Enter your values of A, B, and C or your values for m and b into the appropriate places.

### 5.4.4: Writing the Equation of a Line Passing through a Given Point, Parallel or Perpendicular to a Given Line

You have learned about the special relationship between the slopes of parallel lines and perpendicular lines in subunit 5.3.2. Now that you know how write the equations of lines, you can write the equations of the lines parallel or perpendicular to the given line.

Read this page. Then, complete practice problems 2-10. If you need help, use the "Linear Equations in Standard Form" video embedded in the text for guidance.

Read this page. Then, complete practice problems 3-21. If you need help, use the "Linear Equations in Standard Form" video embedded in the text for guidance.

Complete this exercise. Use the given graph and the scratch pad to draw the line parallel or perpendicular to the given. This will help you to eye the values of its slope and y-intercept. Enter your values of m and b into the appropriate places.

### 5.4.5: Application Problems

Watch these videos and take notes. The first video is an example of writing a linear equation in slope-intercept form to describe a real-world situation (in this case, a cost of a variable number of items). The second video is another example of a linear equation in slope-intercept form that describes a real-world situation. In the third video, the linear equation of a process (motion of the plane) is given. You will learn how to obtain the information about the motion from this equation.

### 5.5: Linear Inequalities in Two Variables

### 5.5.1: Graphing Linear Inequality of Two Variables on the Coordinate Plane

Recall that the solutions of a linear equation or an inequality in two variables are ordered pairs of numbers that make the equation or the inequality a true statement. Both equations and inequalities in two variables have an infinite number of solutions. However, while all solutions of a linear equation belong to one straight line, the solutions of a linear inequality will take up an entire region on a coordinate plane. The following reading explains how to identify this region and show it on the graph.

Read the article, watch the videos embedded in the text, and work through Examples A and B. Then, complete practice problems 1-9. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. You will have to write a linear inequality represented by a given graph. Fill in your answer in the appropriate tab on the left side of the page, and select "Check Answer." If your answer is incorrect, you will be prompted to try again. If it is correct, you can click on "Correct! Next Question" to move on to the next problem.

Complete the exercise, which will help you obtain mastery in graphing and interpreting the solutions of the linear inequalities in two variables. Drag the blue dots to move the line to the correct position and select the appropriate shading and line option (solid or dashed) to represent the given inequality. Then, determine whether the given ordered pairs of numbers are the solutions to this inequality. Once you are done, select "Check Answer." If your graph and answers are incorrect, you will be prompted to try again. If it is correct, you can click on "Correct! Next Question" to move to the next problem.

### 5.5.2: Application Problems

Scroll down and work through Example C and Guided Practice. Then, complete practice problems 14 and 16. Once you have completed the practice problems, check your answers against the answer key.

Complete exercises 3-25. This exercise set will allow you to assess your mastery of most of the concepts covered in Unit 5. Click on the "Show Solution" button next to each problem to check your answer.