2.2 Using the Principles Together

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1 M_BITT77 C_ pp.qxd // : PM Page. USING THE PRINCIPLES TOGETHER Aha! Some equations, like = 7 or x + = x +, have no Solve for x. Assume a, c, m Z. solution and are called contradictions. Other equations, 9. mx =.6m 96. x - + a = a like 7 = 7 or x = x, are true for all numbers and are called identities. Solve each of the following and if an c # cx + c = 7c identity or contradiction is found, state this. a = 7cx a 7. x = x + x x =. ax - a = a x + + x = x 9x = x - x = x + x. If t - 9 =, find t If n + 6 =, find n Lydia makes a calculation and gets an answer of 9. x + = + x On the last step, she multiplies by. when she should have divided by.. What should the 9. x = correct answer be? 9. x = -. Are the equations x = and x = equivalent? 9. x = Why or why not?. Using the Principles Together Applying Both Principles a Combining Like Terms a Clearing Fractions and Decimals EXAMPLE Solve: An important strategy for solving new problems is to find a way to make a new problem look like a problem that we already know how to solve. This is precisely the approach taken in this section. You will find that the last steps of the examples in this section are nearly identical to the steps used for solving the equations of Section.. What is new in this section appears in the early steps of each example. Applying Both Principles The addition and multiplication principles, along with the laws discussed in Chapter, are our tools for solving equations. In this section, we will find that the sequence and manner in which these tools are used is especially important. + x = 7. SOLUTION Were we to evaluate + x, the rules for the order of operations direct us to first multiply by and then add. Because of this, we can isolate x and then x by reversing these operations: We first subtract from both sides and then divide both sides by. Our goal is an equivalent equation of the form x = a. Isolate the x-term. Isolate x. + x = 7 + x - = x = x = x = x = Using the addition principle: subtracting from both sides (adding -) Using a commutative law. Try to perform this step mentally. Using the multiplication principle: dividing both sides by (multiplying by )

2 M_BITT77 C_ pp.qxd // : PM Page 6 6 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING Check: + x = 7 + # 7 + 7? = 7 We use the rules for order of operations: Find the product, #, and then add. STUDY SKILLS Use the Answer Section Carefully EXAMPLE Solve: When using the answers listed at the back of this book, try not to work backward from the answer. If you frequently require two or more attempts to answer an exercise correctly, you probably need to work more carefully and/or reread the section preceding the exercise set. Remember that on quizzes and tests you have only one attempt per problem and no answer section to consult. The solution is. 7 Check: x - 7 =. The solution is 6. SOLUTION In x - 7, we multiply first and then subtract. To reverse these steps, we first add 7 and then either divide by or multiply by. x - 7 = x = + 7 x = # x = # # x = # # x = 6 x - 7 = # =? M Adding 7 to both sides Multiplying both sides by 7 EXAMPLE Solve: - t =. SOLUTION We have - t = - t - = - + -t + - = - r t = - -t = - --t = -- t =. Subtracting from both sides Try to do these steps mentally. Try to go directly to this step. Multiplying both sides by - (Dividing by - would also work.) Check: - t = -? = The solution is. 9 As our skills improve, certain steps can be streamlined.

3 M_BITT77 C_ pp.qxd // : PM Page 7. USING THE PRINCIPLES TOGETHER 7 EXAMPLE Solve: SOLUTION y = -.. We have y = y - 6. = y = y -7. = y =.. Subtracting 6. from both sides Dividing both sides by -7. Check: y = ? = -. The solution is.. Combining Like Terms If like terms appear on the same side of an equation, we combine them and then solve. Should like terms appear on both sides of an equation, we can use the addition principle to rewrite all like terms on one side. EXAMPLE Solve. a) x + x = - b) -x + = -x + 6 c) 6x + - 7x = - x + 7 d) - x + = x - - SOLUTION a) x + x = - 7x = - 7x 7 = - 7 x = - Combining like terms Dividing both sides by 7 The check is left to the student. The solution is -. b) To solve -x + = -x + 6, we must first write only variable terms on one side and only constant terms on the other. This can be done by subtracting from both sides, to get all constant terms on the right, and adding x to both sides, to get all variable terms on the left. Isolate variable terms on one side and constant terms on the other side. -x + = -x + 6 -x + x + = -x + x + 6 Adding x to both sides 7x + = 6 7x + - = 6 - Subtracting from both sides 7x = Combining like terms 7x 7 = 7 Dividing both sides by 7 x = 7 The check is left to the student. The solution is 7.

4 M_BITT77 C_ pp.qxd // : PM Page CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING TECHNOLOGY CONNECTION Most graphing calculators have a TABLE feature that lists the value of a variable expression for different choices of x. For example, to evaluate 6x + - 7x for x =,,, Á, we first use E to enter 6x + - 7x as y. We then use F j to specify TblStart =, Tbl =, and select AUTO twice. By pressing Fn, we can generate a table in which the value of 6x + - 7x is listed for values of x starting at and increasing by ones. 6 X X Y. Create the above table on your graphing calculator. Scroll up and down to extend the table.. Enter - x + 7 as y. Your table should now have three columns.. For what x-value is y the same as y? Compare this with the solution of Example (c). Is this a reliable way to solve equations? Why or why not? c) d) 6x + - 7x = - x + 7 Check: -x + = 7 - x -x + + x = 7 - x + x + x = 7 x = x = x = The student can confirm that checks and is the solution. -x - = x - 7 -x = x r = x + x -6 = x -6 6x + - 7x = - x # # - # x + = x x - = x = x - = x =? The student can confirm that Clearing Fractions and Decimals Combining like terms within each side Adding x to both sides. This is identical to Example. Subtracting from both sides Dividing both sides by Using the distributive law. This is now similar to part (c) above. Combining like terms on each side Adding 7 and x to both sides. This isolates the x-terms on one side and the constant terms on the other. Dividing both sides by This is equivalent to x = -. - checks and is the solution. Equations are generally easier to solve when they do not contain fractions or decimals. The multiplication principle can be used to clear fractions or decimals, as shown here. 9 Clearing Fractions Clearing Decimals x + = A x + B = # x + =.x + 7 =..x + 7 = #. x + 7 = In each case, the resulting equation is equivalent to the original equation, but easier to solve. The easiest way to clear an equation of fractions is to multiply both sides of the equation by the smallest, or least, common denominator of the fractions in the equation.

5 M_BITT77 C_ pp.qxd // : PM Page 9. USING THE PRINCIPLES TOGETHER 9 EXAMPLE 6 Solve: (a) x - 6 = x; (b) SOLUTION x + =. a) We multiply both sides by 6, the least common denominator of and. 6 6a x - 6 b = 6 # x 6 # x - 6 # 6 = 6 # x Multiplying both sides by 6 CAUTION! Be sure the distributive law is used to multiply all the terms by 6. x - = x - = x - = x - = x. Note that the fractions are cleared: 6 # and 6 # =, 6 # 6 =, =. Subtracting x from both sides Dividing both sides by The student can confirm that - checks and is the solution. b) To solve x + =, we can multiply both sides by (or divide by ) to undo the multiplication by on the left side. # x + = # x + = x = x = 6 Multiplying both sides by # ; = and = Subtracting from both sides Dividing both sides by The student can confirm that 6 checks and is the solution. # 6 STUDENT NOTES EXAMPLE 7 Solve: Compare the steps of Examples and 7. Note that although the two approaches differ, they yield the same solution. Whenever you can use two approaches to solve a problem, try to do so, both as a check and as a valuable learning experience. To clear an equation of decimals, we count the greatest number of decimal places in any one number. If the greatest number of decimal places is, we multiply both sides by ; if it is, we multiply by ; and so on. This procedure is the same as multiplying by the least common denominator after converting the decimals to fractions y = -.. SOLUTION The greatest number of decimal places in any one number is two. Multiplying by will clear all decimals y = y = y = - -7y = y = - y = - -7 y =. Multiplying both sides by Using the distributive law Subtracting 6 from both sides Combining like terms Dividing both sides by -7 In Example, the same solution was found without clearing decimals. Finding the same answer in two ways is a good check. The solution is.. 69

6 M_BITT77 C_ pp.qxd // : PM Page 9 9 CHAPTER EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING An Equation-Solving Procedure. Use the multiplication principle to clear any fractions or decimals. (This is optional, but can ease computations. See Examples 6 and 7.). If necessary, use the distributive law to remove parentheses. Then combine like terms on each side. (See Example.). Use the addition principle, as needed, to isolate all variable terms on one side. Then combine like terms. (See Examples 7.). Multiply or divide to solve for the variable, using the multiplication principle. (See Examples 7.). Check all possible solutions in the original equation. (See Examples.). EXERCISE SET For Extra Help i Concept Reinforcement In each of Exercises 6, match the equation with an equivalent equation from the column on the right that could be the next step in finding a solution.. x - = 7 a) 6x - 6 =. x + x = b) x + x =. 6x - = c) x = x = 9 d) x + x = 6 +. x = - x e) 9x = 6. x - = 6 - x f ) x = 9 7 Solve and check. 7. x + 9 =. x - = 9. 6z + = 7. z + = 7. 7t - = 7. 6x - =. x - 9 =. x - 9 =. z + = - 6. x + = = 9t = + t 9. - t = t =. -6z - = -. -7x - = n = n =.. - 7x = x = t - = t - = Aha! x = x = -. - a a. - - = 7 - =. -z - 6z = -. -z + z =. x - 6 = 6x 6. 9y - = y 7. - y = 6 - y x - = 7 + x 7a - = ( - t) = = (x + ) 9 = x - + m - 6 = ( + m) - = 7 r + = 6r + b - = 7b + 6x + = x + y + = y + - x = x - 7x + - x = x - x x - 6 = x + - x + x - 7 = x - - x. y - + y + = 6y + - y

Use the Associative Property of Multiplication to find the product.

Use the Associative Property of Multiplication to find the product. 3-1 1. The Associative Property of Multiplication states factors can be grouped differently and the product remains the same. Changing the grouping of the factors changes the factors that are multiplied