Solving Equations by the Multiplication Property

Transcription

1 2.2 Solving Equations by the Multiplication Property 2.2 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the multiplication property to solve equations. Find the mean for a given set Let s look at a different type of equation. For instance, what if we want to solve an equation like the following? 6 18 Using the addition property of the last section won t help. We will need a second property for solving equations. Rules and Properties: The Multiplication Property of Equality If a b then ac bc where c 0 NOTE Again, as long as you do the same thing to both sides of the equation, the balance is maintained. In words, multiplying both sides of an equation by the same nonzero number gives an equivalent equation. NOTE Do you see why the number cannot be 0? Multiplying by 0 gives 0 0. We have lost the variable! Again, we return to the image of the balance scale. We start with the assumption that a and b have the same weight. a b The multiplication property tells us that the scale will be in balance as long as we have the same number of a weights as we have of b weights. a a a a a b b b b b 1

2 16 CHAPTER 2 EQUATIONS AND INEQUALITIES Let s work through some eamples, using this second rule. Eample 1 Solving Equations by Using the Multiplication Property NOTE 1 6 (6) , or We then have alone on the left, which is what we want. Solve 6 18 Here the variable is multiplied by 6. So we apply the multiplication property and multiply both sides by Keep in mind that we want an equation of the form (6) We can now simplify. 1 or The solution is. To check, replace with : (True) CHECK YOURSELF In Eample 1 we solved the equation by multiplying both sides by the reciprocal of the coefficient of the variable. Eample 2 illustrates a slightly different approach to solving an equation by using the multiplication property. Eample 2 Solving Equations by Using the Multiplication Property Solve NOTE Because division is defined in terms of multiplication, we can also divide both sides of an equation by the same nonzero number. The variable is multiplied by. We divide both sides by to undo that multiplication: 7 Note that the right side reduces to 7. Be careful with the rules for signs. We will leave it to you to check the solution.

3 SOLVING EQUATIONS BY THE MULTIPLICATION PROPERTY SECTION CHECK YOURSELF Eample Solving Equations by Using the Multiplication Property Solve 9 4 In this case, is multiplied by 9, so we divide both sides by 9 to isolate on the left: The solution is 6. To check: ( 9)( 6) (True) CHECK YOURSELF Eample 4 illustrates the use of the multiplication property when fractions appear in an equation. Eample 4 Solving Equations by Using the Multiplication Property (a) Solve 6 Here is divided by. We will use multiplication to isolate To check: (True) This leaves alone on the left because 1 1

4 18 CHAPTER 2 EQUATIONS AND INEQUALITIES (b) Solve 9 ( 9) Because is divided by, multiply both sides by 4 The solution is 4. To check, we replace with 4: (True) The solution is verified. CHECK YOURSELF 4 (a) 7 (b) 4 8 When the variable is multiplied by a fraction that has a numerator other than 1, there are two approaches to finding the solution. Eample Solving Equations by Using Reciprocals Solve 9 One approach is to multiply by as the first step. 9 4 Now we divide by. 4 1 To check: (True)

5 SOLVING EQUATIONS BY THE MULTIPLICATION PROPERTY SECTION A second approach combines the multiplication and division steps and is generally a bit more efficient. We multiply by. NOTE Recall that is the reciprocal of, and the product of a number and its reciprocal is just 1! So So 1, as before. CHECK YOURSELF 2 18 You may sometimes have to simplify an equation before applying the methods of this section. Eample 6 illustrates this property. Eample 6 Combining Like Terms and Solving Equations Solve and check: 40 Using the distributive property, we can combine the like terms on the left to write 8 40 We can now proceed as before The solution is. To check, we return to the original equation. Substituting for yields Divide by The solution is verified. (True)

6 160 CHAPTER 2 EQUATIONS AND INEQUALITIES CHECK YOURSELF An average is a value that is representative of a set of numbers. One kind of average is the mean. Definitions: Mean The mean of a set is the sum of the set divided by the number of elements in the set. The mean is written as (sometimes called -bar ). In mathematical symbols, we say The sum of the set n The number of elements in the set Eample 7 Finding the Mean Find the mean for each set of numbers. (a) 2,,, 4, 7 We begin by finding. 2 ( ) Net we find n. n Remember that n is the number of elements in the set. Finally, we substitute our numbers into the equation. n 1 The mean of the set is. (b) 4, 7, 9,, 6, 2,, 8 First find. ( 4) 7 9 ( ) 6 ( 2) ( ) 8 18 Net find n. n 8 Substitute these numbers into the equation n (or 2.2) 9 The mean of this set is or 2.2 4

7 SOLVING EQUATIONS BY THE MULTIPLICATION PROPERTY SECTION CHECK YOURSELF 7 Find the mean for each set of numbers. (a), 2, 6,, 2 (b) 6, 2,, 8,, 6, 1, Eample 8 Finding the Mean During a week in February the low temperature in Fargo, North Dakota, was recorded each day. The results are presented in the following table. Find both the median and the mean for the set of numbers. M T W Th F Sa Su NOTE You can review the discussion of the median in Section 1.. To find the median we place the numbers in ascending order: The median is the middle value, so the median is 1 degrees. To find the mean, we first find. ( 11) ( 17) ( 1) ( 18) ( 20) ( 2) 20 6 Then, given that n 7, we use the equation for the mean. n The mean is 9.

8 162 CHAPTER 2 EQUATIONS AND INEQUALITIES Which average was more appropriate? There is really no right answer to that question. In this case, the median would probably be preferred by most statisticians. It yields a temperature that was actually the low temperature on Wednesday of that week, so it is more representative of the set of low temperatures. CHECK YOURSELF 8 The low temperatures in Anchorage, Alaska, for one week in January are given in the following table. Compute both the median and the mean low temperature for that week. M T W Th F Sa Su CHECK YOURSELF ANSWERS (a) 21; (b) (a) 2; (b) mean 17, median 22

9 Name 2.2 Eercises Section Date Solve for and check your result. ANSWERS

10 ANSWERS Once again, certain equations involving decimal fractions can be solved by the methods of this section. For instance, to solve we simply use our multiplication property to divide both sides of the equation by 2.. This will isolate on the left as desired. Use this idea to solve each of the following equations for Find the median and the mean of each data set. 4. 2,, 4,, ,, 8, 10, , 1, 2, 4, 6, , 2, 1, 4, 6, , 1, 2, 2,, 7 4, 1, 2,, 6 0. a. b. c. d. e. f. g. h. 49. Average weight. Kareem bought four bags of candy. The weights of the bags were 16 ounces (oz), 21 oz, 18 oz, and 1 oz. Find the median and the mean weight of the bags of candy. 0. Average savings. Jose has savings accounts for each of his five children. They contain $21, $16, $18, $7, and $2. Find the median and the mean amount of money per account. Getting Ready for Section 2. [Section 1.2] Use the distributive property to remove the parentheses in the following epressions. (a) 2( ) (b) (a 4) (c) (2b 1) (d) (p 4) (e) 7( 4) (f) 4( 4) (g) (4 ) (h) (y 2) Answers Mean: 17., Median: 17 oz. a. 2 6 b. a 12 c. 10b d. 9p 12 e f g h. 1y