B Balancing Equations We have learned that in an equation, the epressions on both sides of the equal sign must be equivalent. For eample, + = 1 2 because 7 = 7 6 = 7 because 21 = 21 + + = + 8 + 2 because + 8 = + 8 If the epressions on both sides are not equivalent, it is not an equation. For eample, + 8 because 12 1 8 2 + 2 + 1 because 19 10 2y + y y + 10 6 because y + y + If we add the same number to, or subtract the same number from, both sides of an equation, the epressions on both sides remain equivalent. Similarly, if we multiply or divide both sides of an equation by the same number, the epressions on both sides remain equivalent. For eample, If + 2 = 2, then ( + 2) + = (2 ) + ( + 2) 1 = (2 ) 1 ( + 2) 2 = (2 ) 2 ( + 2) = (2 ) If we add, subtract, multiply, or divide each side by different numbers, the epressions on each side will not be equivalent. For eample, If + 2 = 2, then ( + 2) + (2 ) + ( + 2) 2 (2 ) 1 ( + 2) 2 (2 ) ( + 2) (2 ) 2
Objective: To eplore the properties of equations. 1 Materials: Balance scale labeled with an equal sign in the middle. Two different colored connecting cubes such as yellow and green to represent the variables and constants. Small stickers with to label the yellow (variable) cubes and = to label the middle of the scale. Questions 1. Put yellow cubes together and label them with an, as shown. This shows that =. Place yellow cubes together on the left side and green cubes on the right side of the balance scale, as shown. = 2. Add 2 green cubes to the left side of the balance scale, so we have + 2 on the left side of the scale. =. Net, add 2 green cubes to the right side of the scale. Fill in the with the appropriate number. + 2 = +.. Now, the balance scale shows the equation + 2 =. Take 2 green cubes off the left pan. (a) What happens to the scale? (b) Take 2 green cubes off the right pan now. What happens to the scale? (c) Fill in the with the appropriate number. + 2 2 = + 2. Chapter 9 Equations and inequalities
. Add the same number of green cubes to each side and observe what happens. 6. Add a different number of green cubes to each side and observe what happens. 7. Make a new equation 2 = 6, as shown below. = 8. Add yellow cubes on the left side of the scale, as shown below. = (a) What do you need to do to the right side to balance the scale? (b) Fill in the with the appropriate number. 2 2 = 6. 9. Remove half of the s from the left side of the scale, so that you have 2 on the left side and 12 cubes on the right side. = (a) What do you need to do to the right side to balance the scale? (b) Fill in the with the appropriate number. 2 = 12.
From the results of Class Activity 1, we see that in order to maintain equality, we need to increase or decrease both sides of the equation by the same value. both sides of an equation, it maintains its equality. number, it maintains its equality. Eample Solve the equation + 17 = 6. Algebraically, + 17 = 6 What number plus 17 equals 6? + 17 17 = 6 17 = 19 Check by substitution. Check: 19 + 17 = 6 6 = 6 Subtract 17 from both sides of the equation so that the variable is by itself. Line up the equal signs in your equations and check your answer after you solve for. Method 2 6 17 + 17 = 6 = 6 17 Subtract the part we know, 17, from the total, 6, to find. = 19 REMARKS Total Part = Part Try It! Solve each equation and check you answer. (a) p + 7 = 2 (b) y + = 11. (c) + 1 2 = (d) s + 8. = 66 Chapter 9 Equations and inequalities
Eample Solve the equation 8 = 1. Algebraically, 8 = 1 What number minus 8 equals 1? 8 + 8 = 1 + 8 Check: 22 8 = 1 = 22 1 = 1 8 8 = 1 22 Method 2 8 1 8 1 8 = 1 = 1 + 8 = 22 Add the two parts to find the total. REMARKS Part + Part = Total Try It! Solve each equation and check your answer. (a) n = 12 (b) y 87 = 6 (c) 1 = 1 6 (d) r 0.7 = 1.2 Eample 6 Solve the equation 8 = + 19. Algebraically, 8 = + 19 8 19 = + 19 19 29 = Check: 8 = 29 + 19 = 29 8 = 8 REMARKS When presenting the final solution of an equation, we generally write the variable on the left-hand side like = 29, instead of 29 =. 6
Method 2 8 19 Try It! 6 8 = + 19 + 19 = 8 = 8 19 = 29 Solve each equation and check your answer. (a) = m + 18 (b) = + (c) = y 17 (d) = 2 REMARKS We could also rewrite the equation so the variable is on the left side before solving it. Eample 7 Solve the equation = 2. = 2 = 2 Divide both sides of the equation by. = 1 Check: 1 = 2 2 = 2 Method 2 2 = 2 = 2 = 1 units 2 1 unit 2 = 1 Try It! 7 Solve each equation and check your answer. (a) 6 = 8 (b) n = 27. (c) 2.p = 7. (d) 0.y = 1 Chapter 9 Equations and inequalities 7
Eample 8 Solve =. = = Multiply both sides of the equation by. = 1 Check: 1 = = Method 2 = 1 = = = 1 1 unit units = 1 Try It! 8 Solve each equation and check your answers. n (a) = 28 (b) 7 = 2. y (c) 8 = 17 (d) m = 9. Eample 9 Solve 2 = 16. 2 = 16 2 = 16 Multiply both sides of the equation by to get a whole-number coefficient. 2 = 8 2 Divide both sides of the equation by 2. 2 = 8 2 = 2 8
Method 2 1 1 1 16 2 = 16 2 units 16 1 unit 16 2 = 8 1 units 8 = 2 = 16 2 = 8 = 8 = 2 Method 2 = 16 2 of what number is 16? 2 2 = 2 16 Multiply both sides of the equation by the reciprocal of 2. = 16 2 2 2 = 1 = 2 Check: 2 2 = 16 16 = 16 Thus, = 2 Try It! 9 Solve each equation and check your answers. (a) = 12 (b) y = 28 6 (c) 7 n = 1.8 (d) 8 p = 1 2 Eample 10 Solution Brianna had some money in her savings account. She deposited $87 more and now she has $2. How much money did she have at first? Write an equation and solve the problem. Let $m represent the amount of money Brianna had in her savings account at first. Using a model, Algebraically, m + 87 = 2 m + 87 87 = 2 87 m = 27 She had $27 at first. 2 m + 87 m = 2 87 = 27 m 87 REMARKS The variable represents what we are trying to find. Chapter 9 Equations and inequalities 9
Try It! 10 On Monday, a sith grade class collected some cans for a recycling project. On Tuesday they collected 2 cans. They collected 61 cans in total on these two days. How many cans did they collect on Monday? Write an equation and solve the problem. Eample 11 Solution Try It! 11 Danny had some ribbon. He used 96 cm of it to wrap a present. He has 208 cm of ribbon left. How many centimeters of ribbon did he have at first? Write an equation and solve the problem. Let r cm represent the length of ribbon Danny had at first. Algebraically, Using a model, r 96 = 208 r r 96 + 96 = 208 + 96 r = 0 He had 0 cm of ribbon at first. 96 208 r = 96 + 208 = 0 A bakery made some rolls in the morning. By noon, 16 rolls were sold and 87 rolls were left. Write an equation from the problem and solve it to find how many rolls the bakery made in the morning. REMARKS If you can write the equation without drawing the model, you do not need to draw the model. Eample 12 Solution Abel has one-third as much money as Marvin. If Abel has $6, how much money does Marvin have? Write an equation and solve the problem. Let $ represent the amount of money Marvin has. Then, $ 1 represents Using a model, the money Abel has. 1 = 6 Marvin 1 = 6 = 192 Therefore, Marvin has $192. Abel 6 1 unit 6 units 6 = 192 0
Try It! 12 Jesse has times as many foreign stamps as local stamps. If he has 267 local stamps, how many foreign stamps does he have? Write an equation and solve the problem. Eample 1 Solution Rebecca read times as many pages in her book this week as last week. If she read 1 pages this week, how many pages did she read last week? Write an equation and solve the problem. Let pages represent the number of pages Rebecca read last week. Then, the number of pages she read this week is. Algebraically, = 1 = 1 = 27 Using a model, This Week 1 She read 27 pages last week. Last Week? units 1 1 unit 1 = 27 Try It! 1 Steve spent times as much on a coat as on a pair of shoes. If he spent $90 on the coat, how much did he spend on the pair of shoes? Write an equation and solve the problem. Eample 1 Lily bought a sweater and a coat. The sweater cost as much as the coat. If the sweater cost $6, how much did the coat cost? Using a model, 6 Let $ be the cost of the coat. Then, $ = 6 1 = 6 = 12 = 12 = 60 Thus, = 60. Sweater represents the cost of the sweater. Coat 1 1 1 units $6 1 unit $6 = $12 units $12 = $60 Chapter 9 Equations and inequalities 1
Method 2 = 6 = 6 Multiply both sides of the equation by the reciprocal of. = 60 The coat cost $60. Try It! 1 Ping took as much time as Daniel to solve a puzzle. 8 She took minutes. Write an equation for the problem and solve it to find how much time Daniel took to solve the puzzle. EXERCISE 9.1 BASIC PRACTICE 1. Determine whether = is a solution of each equation. (a) + 12 = 1 (b) 21 = 17 (c) 7 = 11 (d) 8 = + 2. Determine whether y = 12 is a solution of each equation. (a) 7y = 82 (b) 72 = 6y y (c) = (d) 8 = y 2. Determine whether = 8 is a solution of each equation. (a) = + (c) 2. = 18.2 (b) 2 = 1 (d) 6 =. Solve each equation and check your answer. (a) 8 = 10 (b) n 7 = 2 (c) 112 = (d) 10 = 1 FURTHER PRACTICE 6. Determine whether y = 0.6 is a solution of each equation. (a) y = 2 (b).2 = 7y 2 (c) y = 0. (d) 17. y = 18 7. Solve each equation and check your answer. (a) + 8 = 11. (b) n + 10 =. Solve each equation and check your answer. (a) + 19 = 2 (b) y 12 = 1 (c) 61 = n + 7 (d) 26 = p 8 (c) p + 0.7 = 18. (d) 12 = + 1 (e) y 7 = 9 1 (g) 2. = q 1.7 (f) 8 = y 6 2
8. Solve each equation and check your answer. (a) n = 11.2 (b) = 8 (c).w = 7 (e). = 0.7n (g) 8. = 2 (d) (f) MATH@ WORK y = y = 1. Write an equation for each problem and solve the problem. 9. Carlos had some money. After earning $10 from a part-time job, he now has $2. How much money did he have initially? 10. A blue ribbon is twice as long as a red ribbon. If the blue ribbon is 128 cm, how long is the red ribbon? 11. Janice walked to raise money for charity. Peter walked 2 of the distance that Janice walked. If Peter walked 10 miles, how far did Janice walk? 12. A big tank holds times as much water as a small tank. If the big tank holds 7 gallons more than the small tank, how much water does the small tank hold? 1. Mei goes to her gym which is. miles from her home. She cycles some distance of the journey and then runs the rest to get there. If Mei runs 1 of the distance that she cycles, how far does she cycle? 1. Mr. Wong sold cakes at his shop. The number of cakes he sold in May was 18 more than that sold in June. If the total number of cakes sold is 10, how many cakes did he sell in June? 1. In a bo, there are five more quarters than dimes. If the total amount of money in the bo is $8.2, how many dimes are there? BRAIN WORKS 16. The sum of three consecutive odd numbers is 129. Find the product of the three numbers. 17. A tetbook contains the following two problems on algebraic simplification. How would you correct the printing errors to make them correct? (a) 8 + = 11 (b) 11y = 7y 18. A student solved the equation 20 = 16 as follows: 20 = 16 20 = 16 20 = = 20 + = 2 Where did he go wrong? Show one correct way to solve this equation. 19. Three boys want to buy a present for their friend. The present costs $120. Peter and Andrea have $70 altogether. Andrea and Russell have $80 altogether. Peter and Russell have $90 altogether. Do they have enough money to buy the present? Chapter 9 Equations and inequalities
Write in your journal Uses of Inequalities Create a situation that is represented by each of the following inequalities. (a) + 10 (b) 1 Evaluating Epressions and Solving Equations Go online to learn how you can use a spreadsheet like Ecel to evaluate algebraic epressions and to solve equations. For eample, what is the value of 1 y 1 when (a) = 2 and y =, (b) = and y =? A B C 1 y 1 1 y 2 1 2 0. 2 0.1667 0.08 0.0... Step 1: Enter in cell C2 the formula = 1/A2*1/B2 Step 2: Drag the fill handle from C2 to C. How would you use the spreadsheet software to solve these equations? (a) + 12 = 2 (b) = 1 A B C D 1 + 12 2 9 21 11 10 11 Step 1: Enter = A2 + 12 in B2 and fill B and B. Step 2: Enter = *C2 in D2 and fill D and D. 2
Problem Solving corner Eample 1 Beth downloaded 6p e-books. Beth s e-book collection is 2 of Jerry s e-book collection. How many e-books did they download altogether? 6p Beth 1 unit? Jerry? 2 units = 6p 1 unit = 6p 2 = p units = p = 9p 9p + 6p = 1p Method 2 2 units = 6p 1 unit = 6p 2 = p units = p = 1p They downloaded 1p e-books. Try It! 1 June has 6y friends on social media. The number of June s friends is of the number of Chapa s friends. How many friends on social media do they have in all? Chapter 9 Equations and inequalities
Eample 2 Jon was k years old years ago. His mother is times as old as Jon now. How old will Jon s mother be in years? Solution years ago k Jon Now k Jon Mother years later Mother k k k k + k + k + + + + = k + 17 or (k + ) + = k + 12 + = k + 17 In years, Jon s mother will be (k + 17) years old. Try It! 2 Five years ago, Alice was p years old. Alice s grandma is times as old as her now. How old will Alice s grandma be in years?