RWM102: Algebra

Watch the video and take notes. In this video, Sal Khan uses an everyday situation to explain how variables are used to represent numbers.

Read the article and complete practice problems 1-10. This article further explains the meaning and usage of variables. It also provides many examples of using variables to represent quantities that you often encounter in the real world as well as exercises to practice choosing appropriate variables and writing variable expressions to describe situations on your own.

Read this page. This article points out various conventions used in algebra when writing variable expressions, such as always putting a numerical coefficient first and omitting 1 in front of a variable. Use the examples provided as a guide to practice identifying different parts of algebraic expressions. Click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and 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.

Read the brief encyclopedia entry and complete the "Understanding Check" problems. You will receive immediate feedback on whether your response is correct or incorrect when you choose an answer choice. You will be using the terminology introduced here throughout the rest of this course.

Watch the video and take notes. This video gives a very detailed explanation of how to substitute the values of variables in the algebraic expressions. Note how the concept of order of operations is used in some examples. You should be familiar with this concept from your study of arithmetic.

Read this article, which explores various real-world situations where evaluating expressions with one or more variables is required. Then, complete practice problems 1-9 and 26-30. If you need help with problems 1-9, watch the first 2 minutes of the "Variable Expressions" video embedded in the article.

Complete this exercise set. It contains easy problems on evaluating simple algebraic expressions containing only one variable. You will encounter more complex problems later in this subunit. Substitute the given value of x into the given algebraic expression and perform the resultant set of numerical operations.

Complete this exercise set. The algebraic expressions here contain more than one variable and you might also see exponents. You should remember how to handle exponents from your study of arithmetic. Substitute the given values of the variables into the given algebraic expression and perform the resultant set of numerical operations.

Read this page. Order of operations agreement is a convention used by mathematicians to ensure that expressions with many operations are always evaluated the same way, which is consistent with the properties of operations. Because we are used to reading from left to right, it is natural to add and multiply in the same direction, instead of thinking which operations should be performed first. With practice, you will get used to the correct order and will read the expressions accordingly.

Work through examples A and B and Guided Practice problem 1. Complete practice problems 1-7. Make sure to watch the "Order of Operations" video embedded in the text.

Complete this exercise set. It contains basic order of operation problems and will help you assess how well you remember this concept from your study of arithmetic. 

Scroll down to Example C. In this activity, the examples and exercises require an extra step: substituting the values of the variables in the algebraic expressions. Then, they simply become numerical expressions, which can be evaluated by using order of operations agreement.

Work through Example C and Guided Practice problem 2, and then complete practice problems 8-10. Use the "Order of Operations Example" video embedded in the text for guidance. Once you have completed the practice problems, check your answers against the answer key.

Scroll down to the Practice Problems section and complete practice problems 10-25. Watch the "Variable Expressions" video embedded in the text if you need help. Once you have completed the practice problems, check your answers against the answer key.

Review the set of concepts outlined in the "Arithmetic Properties" series of videos, including the commutative and associative laws of addition and multiplication and the identity properties of addition and multiplication. You will study the algebraic applications of these properties in the next assignment.

Read this section through "Sample Set A," and work through the exercises in Practice Set A. The solutions to the practice problems are shown directly below each problem.

You know that changing the order in which two numbers are added does not change the result, and, likewise, changing the order in which two numbers are multiplied does not change the result. Because variables represent numbers, this is true for variables as well. With the help of this reading, you will learn to apply this property to any expression.

Scroll down to the section titled "The Associative Properties." Read this section through "Sample Set B," and work through the exercises in Practice Set B and Practice Set C. The solutions to the practice problems are shown directly below each problem.

The associative property states that several numbers can be added or multiplied in any order. Later in the course, you will be using this property often when you will have to switch terms around in order to simplify algebraic expressions.

Scroll down to the section titled "The Distributive Properties." Read this section through "Sample Set D," and work through the exercises in Practice Set D. The solutions to the practice problems are shown directly below each problem.

The distributive property is another property that will be used extensively in simplifying algebraic expressions.

Read this encyclopedia entry. Note that the like terms have the same variables in the same exponents but might have different numerical coefficients. You will need to recognize like terms in order to add and subtract them.

Watch this video and take notes. In general algebraic expressions, you will need to open the parenthesis and then combine the like terms. Due to commutative and associative properties, you can move the like terms around in order to combine them.

Watch this video and take notes. Dr. Sousa explains a few slightly more complicated examples of simplifying algebraic expressions. Note that the last example in the video contains parenthesis within brackets. According to the order of operations, you need to first remove the inner grouping symbol of parenthesis and then simplify the expression within the brackets.

Attempt several exercises in this section until you feel comfortable with the material. Click on the "Show Solution" link next to each problem to see the correct solution.

Complete this exercise set. It will provide practice using the number properties, particularly commutative and distributive, to simplify simple algebraic expressions. 

Read the following sections: "Define Linear Equations in One Variable" and "Solutions to Linear Equations in One Variable." Then, scroll down to "Exercises," and complete exercises 1-5. Click on the "Show Solution" link for each problem to check your answer.

An important point to take away from this reading is that equation is defined as a statement (containing a variable), which may or may not be true, depending on the value of the variable. To solve an equation means to find all the values of the variable for which the statement is true. In the following subunits, you will focus on finding these values.

Watch this video and take notes. In this video, you will find an ingenious virtual demonstration of a balanced scale representing an equation. This video provides a detailed explanation of why an equation does not change if the same thing is added to (or subtracted from) both sides.

Read this page. After you review the examples, you can use addition property to determine whether two equations are equivalent. Click on the "new problem" button at the end of the article to try a practice problem and check your answer. Continue this process by clicking on "new problem," and try to solve at least 10 practice problems or more, if necessary.

Watch this video and take notes. In this video, the analogy between an equation and a balanced scale is again used to explain why an equation remains the same when both sides are multiplied or divided by the same number or expression.

Read this page. After you review the examples, you can use multiplication property to determine whether two equations are equivalent. Click on the "new problem" button at the end of the article to try a practice problem and check your answer. Continue this process by clicking on "new problem," and try to solve at least 10 practice problems or more, if necessary.

Watch this video. This is an introduction to a hands-on (well, almost) exercise of finding the weight of Spice Man by keeping the scale balanced. To locate the link to the exercise, note that there is a list of videos (marked by a camera symbol) and exercises (marked by a star) on the left-hand side of the page. The link to the current video ("One Step Equation Intuition Exercise Intro") is highlighted in gray, and below this appears the link to the exercise ("One Step Equation Intuition"). Please click on this link, and complete the exercise.

Watch this video and take notes. This video explains how to solve the basic one-step equations you will encounter in algebra and other math courses.

Complete this exercise set. It consists of equations that could be solved by either adding or subtracting a number from both sides in order to isolate the variable. 

Watch these videos and take notes. The first video provides examples of another type of basic equation that can be solved in one step. The second is an example of a one-step equation written in a form x/a = b. Sal Khan uses this example to highlight that division is really the same operation as multiplication. The third contains examples of the equations of the form ax = b, but in some of them a and b are fractions. Instead of dividing by a fraction (a), you can multiply both sides of the equation by its reciprocal.

Read this page, which provides a review and summary of what you have already learned as well as examples of some real-world situations modeled by one-step equations. 

After reading, complete practice problems 1-16. Watch the "One Step Equations" video embedded in the text, if you need help. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise set. Multiply or divide both sides of the equations in order to isolate the variable.

Complete this exercise set to practice solving simple two-step equations.

Watch these videos and take notes. Now that you are familiar with basic equations, the first video introduces an equation that requires more than one step to solve. Note that x is found by performing the operations in the inverse order: first subtraction and then division. The second video contains a very detailed explanation of how to solve two more equations of the form ax + b = c. The third video contains fractions, but it still can be solved as any other equation of the form ax + b = c.

Read the introduction to the examples and work through the examples carefully. Note that these equations are solved here by a method different than the one that was used in the "Algebra: Linear Equations 2" video in subunit 2.2.2.1. Instead of performing operations with fractions, Dr. Burns eliminates the fractions from the equation by multiplying both sides by their common denominator. Click on the "new problem" button at the end of the article to try a practice problem and check your answer. Continue this process by clicking on "new problem," and 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.

Watch these videos and take notes. Now, you are moving on to solving slightly more complicated equations, such as ones that contain variables on both sides as seen in these videos. You can transform this equation into an already familiar form by subtracting a variable expression from both sides.

This page provides practice solving various equation types that have been introduced so far. Please click on the link, and work through the examples. Then, click on the "new problem" button at the end of the page to try a practice problem and check your answer. Continue this process by clicking on "new problem," and solve10 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.

Complete this exercise set to practice solving equations with variables on both sides.

Watch these videos and take notes. The first video will help you to move on to even more complex equations. In the example, you have to remove the parenthesis from the both sides of the equation before attempting to solve it. The second video is an interesting example of an equation with variables on both sides involving fractions. Sal Khan chooses to multiply both sides of the equation by the common denominator and uses distributive property in order to get rid of the fractions.

Read this page to review some more examples of equations with parentheses, and complete practice problems 1-22.  Remember that you need to simplify both sides of equation before solving. Watch the "Multi-Step Equations" video embedded in the text, if you need help. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise set to practice solving equations containing parentheses.

Read the section titled "Conditional Equations, Identities, and Contradictions." Then, complete exercises 26-35. Keep in mind that these equations could have only one solution, infinite solutions, or no solution. Click on the "Show Solution" link next to each problem to check your answer.

Watch this video and take notes. Note that literal equations, just like any others, can be solved by doing the same thing to both sides.

Watch this video and take notes. This video provides another example of solving a literal equation. Note that the steps taken to isolate a variable are very similar to the one in the second example in Dr. Sousa's video.

Scroll down to the section titled "Linear Literal Equations," and work through Example 16. Then, scroll down to "Exercises," and complete exercises 36-40. Click on the "Show Solution" link next to each problem to check your answer.

Complete this exercise set. Solve the problem, and select your answer from the choices given on the right side of the page. Select "Check Answer" to see if you got the answer correct or incorrect. If you get the answer incorrect, it will prompt you to try again. Once you get the answer correct, you can click on "Correct! Next Question" to move on to the next problem.

Watch these videos and take notes. A formula is an equation that expresses a relationship between two or more quantities. When one of these quantities needs to be rewritten in terms of others, the formula becomes a literal equation.

In the first video, the formula for the perimeter of a rectangle (equation used to find a perimeter when its length and width are known) is solved for the width. That is, the result is an equation used to find the width of a rectangle, when its perimeter and length are known. Note that Sal Khan shows two different ways to arrive at the answer.

The formula shown in the second video converts Fahrenheit temperature into Celsius. Solving for Fahrenheit temperature results in a formula that converts Celsius temperature into Fahrenheit. Note that there are two possible approaches to do this, but only one is shown in the video. You might want to try the second approach (distribute 5/9 over the parenthesis) to see which one is more convenient.

Complete exercises 1-11. This exercise set will allow you to assess your mastery of the concepts from Unit 2. Click on the "Show Solution" link next to each problem to check your answer.

Read the section titled "Translating Word to Symbols" through "Sample Set A." Also, copy Table 1 into your notes for future reference. You will use this table in the next assignment.

Complete the exercise set. Use the article in the previous assignment to help you translate these verbal expressions into algebraic expressions.

Watch these videos and take notes. In these videos, Dr. Sousa explains two examples of translating a given real-life situation to algebraic language. As you watch, pay attention to how the keywords such as more and less translate into mathematical operations.

Scroll down to Practice Set A. Using Sample Set A as a guide, complete exercises 1-7 and check your solutions (located below each problem). Developing the skill of translating verbal expressions to mathematical is the first step to solving word problems.

Read the section titled "Five Step Method." This method will guide you through solving problems in this and subsequent subunits of Unit 3. Read Examples 1 and 2 in Sample Set A, and try Exercises 1 and 2 from Practice Set A on your own. The solutions to the practice problems are shown directly below each problem.

• Part 1
• Part 2
• Part 3
• Part 4

On these pages, the author explains how to solve problems involving numbers by translating verbal description of the relationship between the numbers into an equation. Read and work through the solutions to the problems on each page.

Watch these videos and take notes. The first video will introduce you to the word problems involving consecutive integers, or integers that follow one another. Pay attention to the explanation of how to express one consecutive integer in terms of another. Note the difference between the problems: the first one is about consecutive integers, whereas the second one is about consecutive odd integers. The second video is another problem involving consecutive odd integers.

Attempt problem 3 in Part 1, and work through its solution. Again, note how the fact that integers follow one another is expressed algebraically. Then, try problem 6 in Part 2. Compare your solution to the one given by the author. Finally, solve problem 10 in Part 4 and check your answer.

Scroll down to Example 3 in Sample Set A. Review the algebraic method of solving problems involving consecutive, odd, or even integers. Then, complete exercises 4, 34, 36, and 38. Click on the "Show Solutions" link next to each problem to check your answer.

Complete the exercise set. It provides more practice solving problems involving consecutive integers. Keep in mind that after you have solved your equation, you still have to answer the question in the problem.

Watch these videos and take notes. The first video is another example of solving a word problem using a linear equation. In the second video, you may skip the first four minutes (problems 1 and 2) and focus on problems 3 and 4. Note that while these problems are seemingly about different situations, the same basic approach is used.

Watch this video and take notes. This is an example of using algebraic solution to a real-life situation. Watch how each piece of information given about the shelves is translated into algebraic expression and then used to create an equation.

Complete the exercise set. Solving these word problems will help you apply your algebraic and critical thinking skills to various real-life situations.

Watch this video and take notes. In this video, the uniform motion equation is used to find speed and distance. A motion is uniform when the speed, or rate, of the motion is constant. For example, if a car moves with the speed of 30 mph without accelerating or slowing down, then it is in uniform motion. The distance that the car travels during a given time can be found according to the uniform motion equation: distance = rate × time.

Work through the problems on this page on your own and then review the solutions. This page contains examples of two common types of motion problems in which the object travel in the opposite directions, either towards each other or away from each other.

Read this brief example of uniform motion problems, and then attempt the practice problems on the second page. When you have finished, you may check your answers against the answer key.

Read this page. The problems you are going to solve here all have to do with mixtures of two different types of a product, costing a different price. Click on the "new problem" button at the end of the page to try a practice problem, and check your answer. Continue this process by clicking on "new problem," and solve five problems. You can also create a worksheet of five 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.

Watch this video and take notes. This is another example of a mixture problem, where the unit price of the mixture of two products is unknown.

Read this page. Watch the "How to Solve Percent Equations" video embedded in the text to review how to solve basic percent problems by setting up equations, if you need help. Then, complete practice problems 16-30. Use the embedded "Percent Problems" video (skipping the first five minutes) as a guide, if you need help with these problems. Once you have completed the practice problems, check your answers against the answer key.

Complete the exercise set. This will help you assess your proficiency in solving problems involving percents, which you have studied in Arithmetic. You will review the concept and application of percents again in the next assignment.

Watch these videos and take notes. While different in context, algebraically the first problem is similar to mixture problems earlier. Note the steps Sal Khan takes to choose a variable, translate all the information given into algebraic expressions, and set up an equation. The second video is another example of a percent mixture problem. This time, the percent concentration in one part of the mixture is unknown.

This app will guide you one step at a time through setting up the table you need to solve a typical percent mixture problem. This page focuses only on the type of problems in which amounts of both solutions being mixed are unknown. Try to go through at least three problems to achieve proficiency.

This page will help you solve various mixture problems (including the ones involving percents) one step at a time. Try to solve at least five problems or more, if necessary. You can also click on "More information on Money and Mixing Problems" at the bottom of the page to see more examples of mixture and percent mixture problems.

Watch these videos and take notes. In the first problem, two objects (cyclists, in this case) are traveling in opposite directions toward each other. Note how Dr. Sousa uses the table to organize all the information given in the problem.

In the second problem, there is only one traveler, but she travels to another city and then back. Again, a table is useful to organize all the information given. Note that the task here is to find the distance, but the variable chosen to be denoted, x, is the time it takes to complete a one-way trip. You will find that this is a convenient approach in most problems where time is not given directly.

In the third example, two people travel in the same direction and one has to catch up to the other. Note that the key information you need to set up the equation is not given in the problem explicitly: both people will have traveled the same distance by the time one overtakes the other.

This page will guide you through solving various uniform motion problems one step at a time. Try to solve at least five problems or more, if needed. You can also click on "Information on Distance-Rate-Time Problems" at the bottom of the page and scroll down to the section titled "Motion Problems" to see one more solved example of a uniform motion problem.

Read the text until the section titled "The Algebra of Linear Inequalities." This part of the text is an introduction to linear equalities. As you can see from the given examples, inequalities are very similar to equations, except that they state that one expression is smaller (or greater) than another, rather than equal.

Read the brief encyclopedia entry and focus on the table under the title "Graphing One Variable Inequalities." In your notes, make sketches of how four different types of inequalities (in the first four rows of the table) are represented on the number line. Use these notes as a reference throughout the rest of the Unit 4.

Watch this video and take notes. In this video, Sal Khan explains the connection between algebraic inequality statement and its representation on the number line.

Complete the exercise set.

Read this page and watch the videos embedded in the text. After reading the article and working through guided practice examples, complete practice problems 1-12. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise set.

Complete this exercise set.

Read this page. Note that solving inequalities involves the same procedure as solving equations: simplifying both sides, bringing the variable terms to the same side, and isolating the variable. After reading the article and working through guided practice examples, complete the practice problems 1-15. Make sure to watch the embedded "Multi-Step Inequalities" video, if you need help. Once you have completed the practice problems, check your answers against the answer key.

Read the Introduction, and then scroll down to Example 3. Study the solutions for Example 3 (parts a-d), which explain how to translate the verbal statements of inequality into algebraic. Pay attention to the words such as at least or at most, which indicate that greater or equal or less or equal notation should be used.

Watch these videos and take notes. The first video is an example of a word problem that can be solved by using an inequality. The second video is an example of a real-world situation that can be described with an inequality. The third video is another, slightly more complicated example of an application of inequalities.

Complete exercises 13-16 and 24. This exercise set will allow you to assess your mastery of concepts from Unit 4. Click on the "Show Solution" link next to each problem to check your answer.

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.

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.

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.

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.

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.

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

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

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. 

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

Read this page. You can skim through the beginning of the article, as it reviews the concepts you already know. Focus on the definition of a system of equations and its solution as well as the discussion of how many solutions a system of equations can have.

Watch this video and take notes. This video demonstrates how to find out whether an ordered pair of numbers is a solution of a given system of equations.

This article contains practice problems for the concepts you have just learned. Work through Example A, and complete practice problems 9-12. Once you have completed the practice problems, check your answers against the answer key.

Watch this video and take notes. This video demonstrates the method of solving systems of linear equations in two variables by graphing.

Complete this exercise. Graph the equations in the system by using intercepts or slope-intercept method, whichever is more appropriate. Then, determine the coordinates of the point of intersection.

Watch this video and take notes. In this video, a system of equation is solved using substitution.

Read the introduction and section titled "Example (a system with a unique solution)." This reading provides a step-by-step method of solving any system of linear equations in two variables by substitution. However, as you will see in the rest of the subunit 6.1, some systems are better suited for solving by substitution than others.

Complete this exercise. Solve the given system of equations by substitution, then fill in your values of x and y in the appropriate tabs on the right side of the page, and select "Check Answer." 

Complete this exercise. It gives you an opportunity to practice the examples similar to the one you saw in the Khan Academy: video. Fill in your values for x and y in the appropriate places.

Read this page. This reading explains the concepts behind the elimination method and provides different examples of systems that are convenient to solve by elimination.

Watch these videos and take notes. In the first video, you will see an example of a system that is ideal for solving by elimination. The system discussed in the second video can also be solved by elimination, but an extra step is required before one of the variables can be eliminated. In the third example, both equations need to be replaced by equivalent in order to eliminate one of the variables.

Complete this exercise. Solve the given system of equations by elimination.

Read this article and watch the videos embedded in the text. Focus on the summary of the three methods in the "Guidance" section, which highlights when each method is most appropriate. After reading, complete practice problems 1-6. Try to choose the most efficient method for solving each system. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. It provides more practice solving various systems of equations. While Khan Academy suggests a method for solving each system, you might want to think whether this method is most appropriate and possibly choose another one.

Read this page. Also, make sure to watch the "Special Types of Linear Systems" video embedded in the text. Then, complete practice problems 7-24. Keep in mind that you do not have to actually solve a system to classify it as inconsistent or dependent; it is enough to show that slopes of both lines are the same and the y- intercepts are different (in case of inconsistent system) or that the equations in the system are equivalent (in case of dependent system). Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. Use your skills acquired in the previous assignment to determine the number of solutions of each system.

Read this page. This reading reviews the step-by-step process for solving word problems: finding information in the text of the problem, translating verbal statements into mathematical statements, and solving the resultant system of equations. After reading and working through the example, click on "new problem" at the bottom of the page, and solve seven problems. Alternatively, you may generate a worksheet of seven problems by clicking on the button at the bottom of the page titled "Click Here for a Randomly Generated Worksheet and Answers." The solutions will be provided at the end of the worksheet.

Read the entire text, and watch the videos embedded in the text. (You may skip Example B, as you have already seen it in subunit 6.1.6.) This reading discusses a variety of real-world situations which can be described by systems of linear equations. Note that some of these systems can be inconsistent or dependent, and you will learn to interpret what this means in the context of the given situation. After reading, complete practice problems 1-10. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. It contains more word problems that can be solved by setting up and solving a system of equations.

Read the article until the section "Linear Programming - Real-World Systems of Linear Inequalities." Scroll down to the section "Practice Problems," and watch "Systems of Linear Inequalities" video embedded in the text. Then, complete practice problems 6-16. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise. Drag the points to move each line to correct position and select appropriate shading and line option (solid or dashed) to represent each inequality.

Linear programming is a process of analyzing a real-world situation by using a system of several linear inequalities. Read this article and watch the videos embedded in the text. Then, complete practice problems 1-11. Once you have completed the practice problems, check your answers against the answer key.

Scroll down to the section "System of Equations and Inequalities; Counting Methods: Review," and complete the odd-numbered problems for 3-41, but do not complete problem 37. This set of practice problems will allow you to assess your mastery of the concepts in Unit 6. Once you have completed the practice problems, check your answers against the answer key.

Read this article until the section titled "Product of Powers Property." For this subunit on algebraic exponential expressions, complete practice problems 1-13. The definition of a power will be used in the following subunit to derive all of the rules you need to work with exponential expressions. Once you have completed the practice problems, check your answers against the answer key.

Watch this video and take notes. This video points out the difference in exponential expressions such as (-2)4and -24 and explains why the first one equals a positive number while the second one equals a negative.

Read this article, starting with the section titled "Product of Powers Property." Then, complete practice problems 28-43. Watch the "Exponent Properties Involving Products" 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.

Read this page. After reading the article, complete the practice problems 1-26. Once you have completed the practice problems, check your answers against the answer key.

Watch this video and take notes. In this video, Sal Khan explains why the reasoning behind the definition of a negative power as a reciprocal of a positive power. This lecture will also help you understand why the rules of exponents that you have learned so far apply to the negative exponents as well.

Read this page. This article provides a lot of practice applying the definition of negative exponents and simplifying expressions containing negative exponents. Using Sample Sets A, B, and C for guidance, complete the exercises in Practice Sets A, B, and C. Then, complete exercises 33-48. You may access the solutions to the problems by clicking on the "Show Solution" link next to the problem.

Watch these videos and take notes. The first video provides a detailed explanation of an example of taking a power of a product. The second video is another example of simplifying exponential expressions, but this time the example involves a quotient. The third example is more complicated, as it involves raising a product to a negative power.

Read the instructions on how to use this page. Solve five problems correctly at the "Middle" level, and then move up to "Difficult" and correctly solve 5 more problems. Use the "Hint" button if you are not sure where to start. The link "Information about Exponent Rules" at the bottom of the page provides a review and examples of all the relevant rules of exponents.

Scroll down to the Exercises section, and complete exercises 133-143. Click on the "Show Solution" button for each problem to check your answer.

Complete this exercise set. This is another opportunity to review and practice simplifying exponential expressions.

Watch this video and take notes. In this video, Dr. Sousa explains how to multiply monomials using the rules of exponents.

Read the section titled "Guidance," and work through Example A. Then, complete practice problems 1-5. Once you have completed the practice problems, check your answers against the answer key.

Read these examples and practice problems on monomial division. The answers to the practice problems are provided at the end of the slides.

Read the page until the section titled "Classification of Polynomial Equations." Note that while it is not necessary to memorize all the new vocabulary words you will encounter, you should know their meaning, as they will be used often in all the materials in this course from now on. After reading and working through the examples, scroll down to the "Exercises" section and complete exercises 10-34. The solutions to the exercises are shown directly below each problem.

Watch this video and take notes. This short video shows that the total value of a variable number of $20, $10, and $5 bills is a polynomial expression. One can substitute the number of each type of bills into this polynomial to calculate the total amount of money.

Watch this video and take notes. In this video, you will see examples of polynomial expressions of various degrees and number of terms. Then, you will work through two examples of adding and subtracting two polynomials. You will notice that all you have to do to add or subtract polynomials horizontally is open the parentheses (in case of subtraction) and combine like terms. This video also contains an example of applications of polynomials in geometry.

Watch this video and take notes. In this video, adding and subtracting polynomials is performed in both horizontal and vertical formats. Note that in the vertical format, the terms of the same degree (or like terms) are aligned one under another, much like the digits of the same place value are aligned in addition or subtraction of decimals and large numbers.

Watch this video and take notes. In this video, adding and subtracting polynomials is performed in both horizontal and vertical formats. Note that in the vertical format, the terms of the same degree (or like terms) are aligned one under another, much like the digits of the same place value are aligned in addition or subtraction of decimals and large numbers.

Watch this video and take notes. In this video, two polynomials, each containing two variables, are added vertically. Note how like terms are aligned under each other and the rest of the terms are simply rewritten as they cannot be combined.

Scroll down to the section titled "Practice Problems," and complete problems 1-10. Use either horizontal or vertical format of adding/subtracting. Make sure to write the result as a polynomial in standard form. Once you have completed the practice problems, check your answers against the answer key.

Scroll down to Sample Set D, and work through examples 18-21. Then, complete exercises 25-28 from Practice Set D. To check your answers, click on "Show Solution" link next to each problem.

Scroll down to "Example B," and work through Example B, Example C, and Guided Practice. Watch the videos embedded in the text, if you need help with these examples. Then, complete practice problems 6-11. Once you have completed the practice problems, check your answers against the answer key.

Read this page. This reading explains a mnemonic for multiplying binomials, known as FOIL, which stands for First, Outer, Inner, Last, the order in which the terms are multiplied. Practice the basic binomial multiplication problems on this page in order to gain mastery and to be able to do these types of problems quickly.

Click on the "new problem" button at the end of the reading to try a practice problem, and check your answer. Continue this process by clicking on "new problem" five times. You can also create a worksheet of five 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.

Watch this video and take notes. In this video, Sal Khan uses the distributive property to multiply two binomials and then shows that if the order of the binomials in the multiplication problem is switched, then the result remains the same.

Read this page. Here, you will find slightly more complicated binomial multiplication examples. Click on the "new problem" button at the end of the reading to try a practice problem, and check your answer. Continue this process by clicking on "new problem" five times. You can also create a worksheet of five 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.

Watch this video and take notes. In this video, Sal Khan performs the multiplication of a binomial by itself twice: once using FOIL and once using a classic quadratic formula (which he also derives). This will help you understand why this formula always works.

Read this page. This reading provides practice with squaring simple binomials. Click on the "new problem" button at the end of the reading to try a practice problem, and check your answer. Continue this process by clicking on "new problem" five times. You can also create a worksheet of five 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.

Read this page, which contains examples of squaring all kind of binomials. Scroll down to the Practice Set, and complete exercises 1-13. Use the "Special Products of Binomials" 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.

Watch this video and take notes. In this video, Sal Khan performs the multiplication of a binomial by the binomial with same terms but the opposite sign between them twice: once using FOIL and once using a classic quadratic formula (which he also derives). This video will help you understand why this formula always works.

Briefly review this page, which contains examples of applying the difference of two squares formula to all kind of binomials. Scroll down to the Practice Set, and complete exercises 14-23. Use the "Special Products of Binomials" 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 exercises 13-28. This exercise set will allow you to assess your mastery of classification and operations with polynomials (except for division). Once you have completed the practice problems, check your answers against the answer key.

Watch this video and take notes. The video contains an example of multiplication of a binomial of a trinomial. Note the method used to make sure each term of one polynomial is multiplied by each term of another.

Scroll down to the Practice Set, and solve problems 7-21. This set of problems contains exercises involving multiplying polynomials with any number of terms. Some of the problems involve multiplying three polynomials. In this case, multiply any two of the three polynomials first and then multiply the third polynomial by the simplified result. Once you have completed the practice problems, check your answers against the answer key.

Read the section titled "Dividing a Polynomial by a Monomial," work through the examples in Sample Set A, and complete the exercises in Practice Set A. The solutions to the exercises are shown directly below each problem.

Watch this video and take notes. This video shows an example of dividing a polynomial by a monomial.

Watch these videos and take notes. The first video shows an example of dividing a quadratic trinomial by a binomial. Note that the result can be checked by multiplication, just like as the result of the division of real numbers. The second video introduces the method of dividing polynomials known as long division. Sal Khan first tries out this method by dividing a simple binomial by a monomial, which you already know how to do. Then, he generalizes the procedure in order to divide a trinomial by a binomial. Note that he checks his result later by factoring, a procedure you will learn about in Unit 9. You will see more examples of long division in the next two Khan Academy videos as well. The third video shows an example of dividing a cubic four-term polynomial by a binomial. This time, the long division method produces a remainder.

Scroll down to Sample Set B, work through examples 4 and 5, and then work through example 6 in Sample Set C. (Note that example 6 focuses on dividing a polynomial where one of the coefficients is zero.) Then, complete the exercises 6-9 in Practice Set B and exercises 10-13 in Practice Set C. The solutions to the exercises are shown when you click on the "Show Solution" link.

Complete exercises 1-8. Pay careful attention to the directions in each exercise, as they are not all the same. Once you have completed the exercises, check your answers against the answer key.

Read this page. Reading and understanding algebraic expressions, much like translating sentences from a foreign language, is a skill that takes time to develop. Prior to learning how to factor an algebraic expression (that is, to write the expression as a product), it is essential to have fluency in distinguishing which parts of the expressions are multiplied and which are added/subtracted. This reading provides a useful review of how to identify which algebraic expression is a product of several factors and which one is a sum of several terms.

Click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and 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.

Read this page. This set of examples provides practice with identifying common factors, both monomial and binomial, in algebraic expressions. Click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and 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.

Read this page. Note that some examples involve factoring out a common binomial factor.

Try a few simple factoring problems yourself: click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and 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.

Read the section titled "Finding the Greatest Common Binomial Factor." Work through the examples and Guided Practice, and complete practice problems 1-10. Watch the "Polynomial Equations in Factored Form" video embedded in the text, if you need help. Once you have completed the practice problems, check your work against the answer key.

Complete this exercise. Determine if the terms of the given expression have a common factor. If they do, write the factored expression in the tab on the right side of the page. If not, your answer will be the same as the original expression. Select "Check Answer" to see if your answer is correct. If it is 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.

Watch the first 6 minutes of the video and take notes. Note the steps taken to factor a four-term polynomial.

1. Separate the polynomial into two groups of two terms.
   
2. Identify a common factor in each of the group and factor it out.
   
3. Check that the resultant expressions contain a common binomial factor. If they do not, then the polynomial cannot be factored, at least not when the terms are grouped this way.
   
4. Factor out a common binomial factor and rewrite the polynomial as a product of two binomials.
   

Scroll down to the section titled "Factoring by Grouping," and work through Example 3. Then, scroll down to the Practice Set and solve problems 11-15, 17, 18, and 23-27. All of these problems are four-term polynomials that can be factored by grouping. Once you have completed the practice problems, check your answers against the answer key.

Read this page.

Read this page. Note that in one of the examples the trinomial cannot be factored. Then, try factoring trinomials yourself: click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and 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.

Watch this video and take notes. Sal Khan uses both inverse FOIL reasoning and factoring by grouping to factor a quadratic trinomial of the form x² + bx + c with positive c.

Complete this exercise. Factor each trinomial and enter the result in the tab on the right side of the page. Select "Check Answer" to see if your answer is correct. If it is 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.

Read this page. Note that in one of the examples the trinomial cannot be factored. Then, try factoring trinomials yourself: click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and 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.

Watch the first 7 minutes of the first video (introduction and four examples), and take notes. This video reviews the procedure for factoring quadratic trinomials of the form x² + bx + c. The second video shows four examples of factoring polynomials of the form ax² + bx + c.

Read the page until the section titled "Example: 'Factor by Grouping' Method." Try factoring trinomials using this method: click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and solve five problems. You can also create a worksheet of five 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.

While Dr. Burns points out in this reading that listing all trial factors and checking their products can be tedious, some trinomials can be factored fairly quickly using this method. For example, if either a or c or both are prime, their only factors are 1 and itself, and this limits the number of trial factors. Also, if c is positive, both binomial factors will have to contain the same sign, and this limits the number of trial factors as well.

Watch the first five minutes of the first video (introduction and two examples), and take notes. This video explains how to factor the trinomials of the form ax² + bx + c by guessing and checking. The second video shows two more examples of factoring the trinomials of the form ax² + bx + c by guessing and checking.

Revisit this resource. Read the section titled "Example: 'Factor by Grouping' Method." After working through the example, try factoring trinomials using this method yourself: click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem" and solve five problems. You can also create a worksheet of five 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.

This page will guide you through factoring trinomials by grouping one step at a time. Try to factor at least five trinomials or more. You can also click on "Information on the AC method" at the bottom of the page and scroll down to the section titled "The AC Method" to review the procedure for the method, examples, and more exercises.

Work through Example E. In this example, the issue of factoring a trinomial with a = -1 is avoided by factoring (-1) out. Then, the resultant trinomial has the form of x² + bx + c and can be factored as such. Then, work on practice problems 13-16 and 18. Once you have completed the practice problems, check your answers against the answer key.

Complete this exercise to practice factoring various trinomials by grouping.

Watch these videos and take notes. You will see examples of a trinomial containing two variables being factored. Note that the inverse FOIL reasoning can still be applied to determine how the binomial factors are going to look like. Inverse FOIL reasoning, together with the method of trial factors, is again applied to determine how the binomial factors are going to look like.

Complete this exercise. Solve the problem and type your answer in the answer tab.

Read this brief example of prime trinomials, and then attempt the practice problems on the third page. When you have finished, you may check your answers against the answer key.

Read the article until the section titled "Solving Quadratic Polynomial Equations by Factoring." Watch the videos embedded in the text, and work through Guided Practice Examples. Then, complete practice problems 1-8. Once you have completed the practice problems, check your answers against the answer key.

Watch this video and take notes. This video shows how to check whether the given trinomial is a complete square and factor it if it is. (If it is not, factoring by grouping should be attempted.)

Read the article, watch the videos embedded in the text and work through Guided Practice Examples. Then, complete practice problems 1-10. Once you have completed the practice problems, check your answers against the answer key.

Watch this video and take notes. This video shows how to check whether the given binomial is a difference of two squares and factor it if it is.

Complete this exercise to practice factoring simple Difference of Two Squares binomials.

Complete this exercise. It contains slightly more complicated Difference of Two Squares binomials.

Watch these videos and take notes. In these videos, the sum of two cubes formula is proven by using polynomial division and there is an example of factoring a binomial that is a difference of two cubes.

Watch this video and take notes. In this video, Sal Khan shows why the sum of two cubes formula is true (in a different way than in Dr. Sousa's video) and uses it to factor a binomial.

Read the section titled "Guidance," and work through Example A. Scroll down to the Guided Practice, and work through the examples. Then, complete practice problems 1-10. Once you have completed the practice problems, check your answers against the answer key.

Watch these videos and take notes. The first video shows an example of a trinomial that requires two factoring steps in order to be factored completely: factoring out a common monomial factor and factoring by grouping. The second shows an example of a binomial that requires two factoring steps in order to be factored completely: factoring out a common monomial factor and using the difference of two cubes formula.

Watch these videos and take notes. In the first example, the difference of two squares formula is applied twice in order to factor the expression completely: once to the original polynomial, and then to the new binomial factor. The second is an example of an interesting polynomial that can be attempted to be factored either by using the difference of two squares or difference of two cubes formulas. Whichever one you choose, the resultant factors can still be factored further using another special binomial factoring formula. In this video, the difference of two squares formula is used first. As an exercise, try an alternative method (e.g. applying difference of two cubes first) and try to show that the results will in fact be the same.

Complete exercises 3-13. This exercise set will allow you to assess your mastery of factoring polynomials. Click on the "Show Solution" link next to each problem to check your answer.

Complete this exercise. These trinomials require two steps to be factored completely - factoring out the greatest common factor and then factoring the remaining trinomial.

Complete this exercise. It contains binomials that require two steps to be factored - factoring out the greatest common factor and using the Difference of Two Squares formula.

Read these examples and practice problems on factoring a polynomial. The answers to the practice problems are provided at the end of the slides.

Watch the first five minutes of the video, and take notes. In this video, the sum of two cubes formula is proven by using polynomial division. Also, there is an example of factoring a binomial that is a difference of two cubes.

Watch this video and take notes. In this video, you will see a few examples solved by various methods of factoring.

Work through the examples. Then, click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and 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.

Work through the examples. Then, click on the "new problem" button at the end of the page to try a practice problem. After answering the problem, click on the "check your answer" button. Continue this process by clicking on "new problem," and 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.

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