Unit 2: Linear Equations
You probably use equations in everyday life without even realizing it. Calculating a unit price to figure out which brand is cheaper when making a purchase, converting inches into feet, estimating how much time it would take you to drive to your destination at a certain speed all involve solving equations mentally. In this unit, you will learn formal procedures for solving equations. You can probably recall the basic rules from the math courses you have taken in the past. In this unit, you will review these rules while focusing on the formal logical definition of equation as a statement and its solution as a number that makes this statement true. You will apply the skills from Unit 1 to simplify the sides of an equation before attempting to solve it. You will also work with equations that contain more than one variable, and you will learn that because variables always represent numbers, you can use the same rules to find the specific variable you are looking for.
Completing this unit should take you approximately 14 hours.
Upon successful completion of this unit, you will be able to:
- determine whether a given real number is a solution of an equation;
- simplify equations using addition and multiplication properties;
- find solution of a given linear equation with one variable;
- determine the number of solutions of a given linear equation in one variable;
- classify linear equations in one variable according to the number of solutions;
- solve a literal equation for the given variable; and
- rearrange formulas to isolate a quantity of interest.
2.1: Introduction to Equations
2.1.1: Definition of an Equation and a Solution of an Equation
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.
2.1.2: Addition/Subtraction Property of Equations
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.
2.1.3: Multiplication/Division Property of Equations
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.
2.2: Solving Equations
2.2.1: Basic One-Step Equations
2.2.1.1: Equations of the Form x + a = b and x - a = b
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.
2.2.1.2: Equation of the Form ax = b and x/a = b
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.
2.2.2: Equations with Variables on One Side
2.2.2.1: Equations of the Form ax + b = c
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.
2.2.2.2: Equations of the Form ax + b = c Containing Fractions
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.
2.2.3: Equations of the Form ax + b = cx + d
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.
2.2.4: Equations Containing Parentheses
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.
2.2.5: Classifying Equations According to the Number of Solutions: Identities and Contradictions
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.
2.3: Literal Equations
2.3.1: Solving Literal Equation for One of the Variables
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.
2.3.2: Formulas
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.