1.8 INTRODUCTION TO SOLVING LINEAR EQUATIONS

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1 1.8 INTRODUCTION TO SOLVING LINEAR EQUATIONS Combining Like Terms In algebra we often deal with terms such as 4y or 7x. What do we mean by terms? A term is a number, a variable, or a product of a number and one or more variables. Terms are separated from other terms in an expression by a + sign or a - sign. Often a term has a number factor and a variable factor. The number factor is called the coefficient. numerical part of term 7x variable part of term A term that has no variable is called a constant term, and a term that has a variable is called a variable term. Student Learning Objectives After studying this section, you will be able to: Combine like terms. Translate English statements into equations. Solve equations using basic arithmetic facts. Translate and solve equations. 9+3n+4x constant term variable terms What do the expressions 3n and 4x mean? 3n is the term that represents the sum n + n + n, and 4x is the term that represents the sum x + x + x + x. As we saw earlier, 3n and 4x also indicate multiplication: 3 times n and 4 times x. n + n + n T 3n We count three n s added. x + x + x + x T 4x We count four x s added. EXAMPLE 1 (a) Two y s (c) Seven Write a term that represents each of the following. (b) a + a + a + a (d) One x (a) Two y s = 2y (b) a + a + a + a = 4a (c) Seven = 7 (d) One x = 1x or x Practice Problem 1 (a) Four n s (c) Eight Write a term that represents each of the following. (b) y + y + y (d) One y NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 We see many examples of adding and subtracting quantities that are like quantities, as shown in the following example. 3 feet + 7 feet = 10 feet 7 trucks - 2 trucks = 5 trucks However, we cannot combine things that are not the same: 7 trucks - 4 feet 1cannot be done!2 Similarly, in algebra we cannot combine terms that are not like terms. Like terms are terms that have identical variable parts. For example, in the expression 8x + 6b + 2x, the terms 8x and 2x are called like terms since they have the same variable parts. They are both counting x s. 69

2 70 Chapter 1 Whole Numbers and Introduction to Algebra Expression Like Terms 8x+6b+2x 8x 2x The variable parts are the same. There are no like terms for 6b, since none of the terms have exactly the same variable part as 6b. EXAMPLE 2 Identify the like terms. 7ab + 4a + 2ab + 3y There are no terms like 4a; none have the exact variable part, a. 7ab+4a+2ab+3y There are no terms like 3y; none have the exact variable part, y. 7ab and 2ab are like terms; the variable parts, ab, are the same. Practice Problem 2 Identify the like terms. 2mn + 5y + 4mn + 6n The numerical part of a term is called the coefficient of a term. The coefficient tells you how many you have of whatever variable follows. To combine like terms, we either add or subtract the coefficients of like terms. COMBINING LIKE TERMS To combine like terms, add or subtract the numerical coefficients of like terms. The variable parts stay the same. 6x + 4x = 10x 8y - 2y = 6y EXAMPLE 3 Identify like terms, then combine like terms. (a) 3x + 7y + 2x (b) 9m - m - 8 (c) 4xy + 8y + 2xy There are no terms like 7y; none have the exact variable part. (a) 3x + 7y + 2x = 1 3x + 2x 2 + 7y = x + 7y 3x + 7y + 2x = 5x + 7y We identify and group like terms. Three x s plus two x s can be restated as x s; three plus two x s x = 5x (b) 9m - m - 8 = 9m - 1m - 8 Write the numerical coefficient 1. = 9 m - 1 m - 8 Think: nine m s minus one m equals 8 m s. 9m - m - 8 = 8 m - 8 Note that the term m does not have a visible numerical coefficient. We can write 1 as the numerical coefficient since 1m = 1 # m = m.

3 (c) 4xy + 8y + 2xy = 14xy + 2xy2 + 8y = xy + 8y 4xy + 8y + 2xy = 6xy + 8y Section 1.8 Introduction to Solving Linear Equations 71 We identify and group like terms. Add the numerical coefficients of like terms. We write 8y as a separate term. We cannot combine it with 6xy since the variable parts are not the same. Practice Problem 3 Identify like terms, then combine like terms. (a) 2ab + 4a + 3ab (c) 7x + 3y + 3z (b) 4y + 5x + y + x EXAMPLE 4 Write the perimeter of the rectangular figure as an algebraic expression and simplify. 4a 7b 2a 3b Since the figure is a rectangle, opposites sides are equal. 2a 3b 4a 7b 4a 7b 2a 3b We add all sides to find the perimeter: 12a + 3b2 + 14a + 7b2 + 12a + 3b2 + 14a + 7b2 = 12a + 2a + 4a + 4a2 + 13b + 3b + 7b + 7b2 = 12a + 20b We must combine like terms. We use the associative and commutative properties to change the order of addition and regroup. We combine like terms. The algebraic expression for the perimeter is 12a + 20b. Practice Problem 4 Write the perimeter of the triangular figure in the margin as an algebraic expression and simplify. 2x y 3x 2y x 4y Translating English Statements into Equations Two expressions separated by an equals sign is called an equation. When we use an equals sign 1=2, we are indicating that two expressions are equal in value. 2+6=8 The value of this expression is 8. The value of this expression is 8. Some English phrases for the symbol = are is is the same as equals is equal to the result is

4 72 Chapter 1 Whole Numbers and Introduction to Algebra EXAMPLE 5 Translate each English sentence into an equation. (a) Three subtracted from what number is equal to ten? (b) Five times what number is the same as thirty-five? (c) Kari s savings decreased by $100 equals $500. (a) Three subtracted from what number is equal to ten? n - 3 = 10 c c (b) Five times what number is the same as thirty-five? T T T T T 5 # n = 35 5n = 35 (c) We let x represent Kari s savings, the unknown value. Kari s savings decreased by $100 equals $500. T T T T T x - $100 = $500 x - $100 = $500 c NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 Practice Problem 5 Translate each English sentence into an equation. (a) Four times what number is the same as seven? (b) Three subtracted from what number is equal to nine? (c) The number of baseball cards in a collection plus 20 new cards equals 75 cards. Solving Equations Using Basic Arithmetic Facts Suppose we ask the question Three plus what number is equal to nine? The answer to this question is 6, since three plus six is equal to nine. The number 6 is called the solution to the equation 3 + x = 9 and is written x = 6: the value of x is 6. In other words, an equation is like a question, and the solution is the answer to this question. English Phrase Math Symbols Question Equation Three plus what number is equal to nine? Answer to the Question 3 + x = 9 Three plus six is equal to nine. x = 6 The solution to an equation must make the equation a true statement. For example, if 6 is a solution to 3 + x = 9, we must get a true statement when we evaluate the equation for x = x = = 9 9 = 9 We replace the variable with 6, then simplify. We get a true statement. To solve an equation we must find a value for the variable in the equation that makes the equation a true statement.

5 Section 1.8 Introduction to Solving Linear Equations 73 EXAMPLE 6 Is 2 a solution to 6 - x = 9? If 2 is a solution to 6 - x = 9, when we replace x with the value x = 2 we will get a true statement. 6 - x = Six minus what number equals nine? Replace the variable with 2 and simplify. This is a false statement. Since 4 = 9 is not a true statement, 2 is not a solution to 6 - x = 9. Practice Problem 6 Is 5 a solution to x + 8 = 11? NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 EXAMPLE 7 Solve the equation 3 + n = 10 and check your answer. To solve the equation 3 + n = 10, we answer this question: Three plus what number is equal to ten? Using addition facts we see that the answer, or solution, is 7. To check the solution, we replace n with the value n = 7 and verify that we get a true statement. Check: 3 + n = = 10 Write the equation. Replace the variable with 7 and simplify. Verify that we get a true statement. Since we get a true statement, the solution to 3 + n = 10 is 7 and is written n = 7. Practice Problem 7 Solve the equation 4 + n = 9 and check your answer. EXAMPLE 8 Solve the equation 9n = 45 and check your answer. To solve the equation 9n = 45, we answer this question: Nine times what number equals forty-five? The answer or solution is 5 and is written n = 5. To check the answer, we replace n with the value n = 5 and verify that we get a true statement. Check: 9 n = Replace the variable with 5 and simplify. 45 = 45 Verify that this is a true statement. Thus the solution to 9n = 45 is 5 and is written n = 5. Practice Problem 8 Solve the equation 6x = 48 and check your answer.

6 74 Chapter 1 Whole Numbers and Introduction to Algebra 6 EXAMPLE 9 Solve the equation and check your answer. x = 3 6 Six divided by what number equals 3? x = 3 Using division facts, we see that the answer or solution is 2 and is written x = 2. Check: 6 x = 3 : Replace the variable with 2 and simplify. 3 = 3 Verify that this is a true statement. Thus x = 2 is the solution. x Practice Problem 9 Solve the equation 4 = 2 and check your answer. Sometimes, we must first use the associative and commutative properties to simplify an equation and then find the solution. EXAMPLE 10 Simplify using the associative and commutative properties and then find the solution to the equation 15 + n2 + 1 = 9. First we simplify. Next we solve n + 6 = n2 + 1 = 9 1n = 9 n = 9 n + 6 = 9 n + 6 = 9 n = 3 Commutative property Associative property Simplify. What number plus 6 is equal to 9? We leave the check to the student. The solution to the equation is 3 and is written n = 3. Practice Problem 10 Simplify using the associative and commutative properties and then find the solution to the equation 13 + x2 + 1 = 7. We may need to combine like terms before we solve an equation. EXAMPLE 11 Simplify by combining like terms and then find the solution to the equation n + 5n = 18. n + 5n = 18 Write the equation. 1 n + 5n = 18 Write n as 1n n = 18 Add numerical coefficients of like terms. 6n = 18 Think: Six times what number equals eighteen? 3. n = 3

7 Section 1.8 Introduction to Solving Linear Equations 75 Practice Problem 11 Simplify by combining like terms and then find the solution to the equation n + 3n = 20. NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 Translating and Solving Equations In many real-life applications we must translate an English statement into an equation and then solve the equation. EXAMPLE 12 eighteen? Translate, then solve. Double what number is equal to Double what number is equal to eighteen? T T T T 2 # n = 18 n = 9 Translate. Use multiplication facts to find n. Practice Problem12 equal to twenty? Translate, then solve. What number times five is Understanding the Concept Evaluate or Solve? Do you know the difference between evaluating the expression 8x when x is 3 and solving the equation 8x = 16? Evaluate an expression. We replace the variable in the expression with the given number and then perform the calculation(s). Evaluate 8x when x is 3. 8 # 3 = 24 Solve an equation. We find the value of the variable that makes the equation a true statement that is, the solution to the equation. Solve: 8x = 16. x = 2 We can illustrate this idea with the following situations. 1. Evaluating (a) Fact. You are given directions to the Lido Movie Theater. (b) Evaluate. You follow these directions to the movie theater. 2. Solving (a) Fact. You know the address of the theater. (b) Solve. You must find the directions yourself. In summary, an equation has an equals sign, and an expression does not. We find the solutions to equations, and we evaluate expressions as directed. Exercise 1. Can you think of other real-life situations that illustrate the difference between evaluating and solving?

8 76 Chapter 1 Whole Numbers and Introduction to Algebra Improving Your Test-Taking Skills Step 1 Write key facts on your test. As soon as you get your test, find a blank area to write down any important strategies, formulas, or key facts. An ideal place to write these facts is on the blank back page of the test. If there are no blank pages or areas on the test, be sure to ask the instructor if you can have a blank piece of paper for this purpose. Having this information easily accessible should lessen your anxiety and help you focus on the type of problem-solving techniques you need. Step 2 Scan the test and work problems that are easy first. Quickly glance at each question on the test, placing an * beside the ones you feel confident you can complete. Then complete these problems first. This will help build your confidence. Step 3 Keep track of the time as you complete the rest of the test. Determine how much time is left so you can plan the strategy for the rest of the test. This plan should include determining how many minutes you should spend on each of the remaining unanswered questions so that you can finish the test in the time that is left. For example, if there are 10 questions remaining on the test and 30 minutes left, you should try to spend no more than 3 minutes on each of the remaining questions. Step 4 Complete the rest of the problems on the test. Complete the remaining problems on the test, starting with the ones you feel most confident about. If you get stuck on a problem, stop working on it and move on to another one. Do not spend too much time trying to complete one problem; leave it and move on! Step 5 Relax periodically. If you start to feel anxious at any time during the test, take a few moments to relax. Close your eyes, place yourself in a comfortable position in your chair, breathe deeply, and take a moment to think about something pleasant. Next, think positive thoughts such as I will answer the question to the best of my ability and will not worry about what I have forgotten or do not understand. Instead I will show that I can master what I do understand. You may think that taking a few minutes away from the test to relax is wasting time. This is not true. You will perform better if you are relaxed. Step 6 Revisit the problems you are not sure of or did not complete. Try to rework the problems that you struggled with earlier. You may recall how to complete these problems once you have completed the majority of the test. Step 7 Review the entire test to check for careless errors. Take whatever time is left to review all your work. Check for careless errors and be sure that you have followed all directions properly.