Solving Linear Equations: Chapter 8 PDF

CHAPTER 8 Solving Linear Equations A balanced diet is one of the keys to good health, physical and mental development, and an active life. Making healthy food choices requires knowledge of your nutritional needs and of the nutrients found in foods. There are resources to help you...

CHAPTER 8 Solving Linear Equations A balanced diet is one of the keys to good health, physical and mental development, and an active life. Making healthy food choices requires knowledge of your nutritional needs and of the nutrients found in foods. There are resources to help you. An example is Canada’s Food Guide. It stresses the importance of eating a variety of food such as vegetables, fruits, and whole grains. It is also important to control your intake of fat, sugar, and salt. Web Link To get a copy of Canada’s Food Guide, go to www.mathlinks9.ca and follow the links. The links also provide useful resources on how to use the food guide. These resources include “Eating Well with Canada’s Food Guide” and “Eating Well with Canada’s Food Guide—First Nations, Inuit and Métis.” What You Will Learn to model problems using linear equations to solve problems using linear equations 288 Chapter 8 NEL  Key Words equation constant variable opposite operation numerical coefficient distributive property Literacy Link A concept map can help you visually organize your understanding of math concepts. Create a concept map in your math journal or notebook. Make each oval large enough to write in. Leave enough space to attach additional ovals to each strategy shown. As you work through the chapter, complete the concept map. For each strategy develop an example list the steps for solving the equation Discuss your strategies with a classmate. You may wish to add to or correct what you have written. with multiplication with two and division operations Solving Equations with grouping with variables symbols on both sides NEL Chapter 8 289  FOLDABLES TM Study Tool Making the Foldable Step 4 Materials Use half a sheet of 8.5 × 11 paper to create a pocket for sheet of 11 × 17 paper storing Key Words and examples of linear equations. ruler four sheets of 8.5 × 11 paper Step 5 scissors Staple the three booklets and the pocket you made stapler into the Foldable from Step 1 as shown. index cards Solve Check Check Solve Solve Step 1 ax = b ax = b + cx ax + b = a(bx + c) = ax + b cx + d d(ex + f ) =c Fold the long side of a sheet Check of 11 × 17 paper in half. Pinch _a = b _x + b it at the midpoint. Fold the x a =c outer edges to meet at the midpoint. Label it as shown. Key Words _x = b a(x + b ) a =c Step 2 Solve Check Solve Check Using the Foldable ax = b ax + b As you work through Chapter 8, use the shutter-fold Fold the short side of a sheet =c of 8.5 × 11 paper in half. booklets on the left and right panels to check your Pinch it at the midpoint. Fold understanding of the concepts. Write an equation _a = b _x + b the outer edges to meet at x a below the left tab, in the form indicated. Write the =c the midpoint. Fold in three solution to the equation under the right tab. Use the the opposite way. Make four centre shutterfold in a similar manner, writing the _x = b a(x + b ) equation under the top tab and the solution under cuts as shown through one a =c thickness of paper, forming the bottom tab. six tabs. Use the back of the Foldable to record your ideas for Repeat Step 2 to make another shutter-fold booklet. the Math Link: Wrap It Up! Label them as shown. On the front of the Foldable, make notes under the Step 3 heading What I Need to Work On. Check off each item as you deal with it. Fold the long side of a sheet of 8.5 × 11 paper in half. Pinch it at the midpoint. Fold the outer edges of the paper to meet at the midpoint. Fold in three the opposite way. Make four cuts through one thickness of paper, forming six tabs. Label the tabs as shown. Solve ax = b + cx ax + b = a(bx + c) = cx + d d(ex + f ) Check 290 Chapter 8 NEL Math Link Solve Problems Involving Nutrition Some problems that involve nutrition can be modelled using linear equations. In this chapter, you will use linear equations of different forms to model problems involving various foods. 1. Model each problem with an equation. Then, solve the equation. Share your method with your classmates. Web Link a) The mass of carbohydrate in a medium-sized peach is 5 g less than the mass To learn more about of carbohydrate in a medium-sized orange. The peach contains 10 g of nutrition, healthy carbohydrate. What mass of carbohydrate is in the orange? eating, and the b) Half a pink grapefruit contains 47 mg of vitamin C. What mass of vitamin C nutritional values of does a whole pink grapefruit contain? different foods, go to c) One litre of skim milk contains 1280 mg of calcium. What is the mass of www.mathlinks9.ca and calcium in one 250-mL serving of skim milk? follow the links d) A 250-mL serving of baked beans in tomato sauce contains 11 g of fibre. This mass of fibre is 1 g more than the mass of fibre in two 85-g servings of whole wheat pasta. What mass of fibre is in one 85-g serving of whole wheat pasta? e) The mass of potassium in a medium-sized apple, including the skin, is about 160 mg. This mass is 10 mg more than one third of the mass of potassium in an average-sized banana. What mass of potassium is in the banana? f) One serving of a snack contains 250 mL of dried apricots and 125 g of low-fat, plain yogurt. Three servings of this snack contain 36 g of protein. If 125 g of yogurt provides 7 g of protein, how much protein is in 250 mL of dried apricots? 2. a) Develop two different problems involving nutrition that can be modelled using linear equations. Use the Internet or the library to research the nutritional information. Make sure you can solve the problems you create. b) Ask a classmate to solve your problems. Verify your classmate’s solutions. NEL Math Link 291  8.1 Solving Equations: x ̲ a ̲ ax = b, a = b, x = b Focus on… After this lesson, you will be able to… Steve Nash is arguably the most successful model problems basketball player that Canada has produced. He with linear equations grew up in Victoria, British Columbia. In high that can be solved school, he led his team to the provincial using multiplication championship and was the province’s and division player of the year. He has since solve linear become one of the top players in the equations with National Basketball Association rational numbers (NBA). using multiplication and division One year, Steve scored 407 points for the Phoenix Suns in 20 playoff games. If the equation 20x = 407 represents this situation, what does x represent? What operation could you use to Did You Know? determine the value of x? A Canadian, Dr. James Naismith, invented the game of basketball in 1891. At the time, he Explore Equations Involving was teaching in Springfield, Multiplication and Division Massachusetts. 1. a) Each paper clip represents the variable x. Each circle represents a cup viewed from above. How does the diagram model the linear equation 2x = 0.12? Web Link 2009 CA What other combination N A D NADA = A To learn more about of coins could you use to A C 2009 10 CENTS Steve Nash’s life, his 2009 CA NADA represent 0.12? career, and his work in communities, go to b) How does the diagram model the solution to the equation in part a)? www.mathlinks9.ca What is the solution? and follow the links. 2009 CA N A D NADA = A A C 2009 10 CENTS Materials coins 2009 CA NADA paper cups or small containers CENT CENT paper clips S S 5 5 2009 CA N A D NADA = A 2009 2009 A C 2009 10 CENTS 2009 2009 CA CA NADA NADA 2009 CA NADA c) Explain how the second part of the diagram in part b) can also model the equation 0.06y = 0.12. What is the solution? Explain. 292 Chapter 8 NEL  2. Work with a partner to explore how to model the solutions to the following equations using manipulatives or diagrams. Share your Literacy Link models with other classmates. An equation is a statement that two a) 3x = 0.6 mathematical b) 0.05y = 0.25 expressions have the same value. Examples of equations include 3. a) How does reversing the second part of the diagram from #1b) y n = 0.06? 3x = -2, __ = 1, and model the solution to the equation __ 2 2 z = -2.7. CENT CENT In the equation 1.2d + 3.5 = -1.6, S S 5 5 2009 CA NADA N A D = A 2009 2009 A C 2009 10 CENTS 2009 2009 d is the variable, which represents an CA CA 2009 CA NADA NADA NADA unknown number b) What is the solution? Explain. 1.2 is the numerical c) Describe how the diagram in part a) can also model the solution to coefficient, which 0.12 = 0.06. What is the solution? Explain. the equation _____ multiplies the k variable 3.5 and -1.6 are constants 4. Work with a partner to explore how to model the solutions to the following equations using manipulatives or diagrams. Share your models with other classmates. x a) __ = 0.05 3 0.33 b) _____ = 0.11 c Reflect and Check 5. a) How can you model solutions to equations of x = b, and __ the form ax = b, __ a = b using a x manipulatives or diagrams? b) Think of other ways to model the solutions. Explain how you would use them. Digital rights not available. 6. a) When a basketball player takes the ball away from an opposing player, it is called a steal. In his first nine seasons in the NBA, Steve Nash averaged 0.8 steals per game. Write and solve an equation that can be used to determine how many games it took, on average, for Steve to achieve four steals. b) Use at least one other method to solve the problem. Share your solutions with your classmates. x a NEL 8.1 Solving Equations: ax = b, __ __ a = b, x = b 293  Link the Ideas Example 1: Solve One-Step Equations With Fractions Solve each equation. 3 a) 2x = __ m 2 b) __ = -__ 1 1 c) -2__k = -3__ 4 3 5 2 2 Solution 3 a) You can solve the equation 2x = __ using a diagram or algebraically. 4 Strategies Method 1: Use a Diagram 3 on a number line. Model the equation 2x = __ Draw a Diagram 4 + _3 4 0 +1 2x The length of the curly bracket represents 2x, so half of this length represents x. + _3 4 Why were the divisions 0 +1 on the number line changed from quarters 2x to eighths? x x 3. The diagram shows that x = __ 8 Method 2: Solve Algebraically Literacy Link Solve by applying the opposite operation. An opposite operation 3 2x = __ “undoes” another 4 Why do you divide both sides by 2? operation. Examples of 3 __ 2x ÷ 2 = ÷ 2 opposite operations 4 3 are: To divide __ by 2, why do you x= ×3 __ 1 __ 4 1? multiply by __ addition and 4 2 2 subtraction 3 = __ multiplication and 8 division Opposite operations Check: are also called inverse Left Side = 2x 3 Right Side = __ operations. 4 ( ) = 2 __3 8 6 = __ 8 3 = __ 4 Left Side = Right Side 3 , is correct. The solution, x = __ 8 294 Chapter 8 NEL  m 2 b) You can solve the equation __ = -__ using a diagram or algebraically. 3 5 Method 1: Use a Diagram 2 on a number line. m = -__ Model the equation __ 3 5 2 -_ 5 -2 -1 0 m __ 3 m , so use three of these to The length of the curly bracket represents __ represent m. 3 2 -_ 5 -2 -1 0 m __ m __ m __ 3 3 3 m 6 , or -1__ The diagram shows that m = -__ 1. 5 5 Method 2: Solve Algebraically Solve by applying the opposite operation. m = -__ __ 2 What is the opposite 3 5 operation of dividing by 3? 3 × __ 3 ( ) m = 3 × -__ 2 5 3 × ___ m = __ -2 1 5 ( ) 2 ≈ 3 × -__ 3 × -__ 5 1 2( ) -6 6 or -1__ = ___ or -__ 1 1 ≈ -1__ 5 2 5 5 Check: 1 m 2 -1__ is close to the estimate, Left Side = __ Right Side = -__ 5 3 5 so this answer is reasonable. -__6 5 = ____ 3 6÷3 = -__ 5 1 -6 × __ = ___ 5 3 -6 = ___ 15 2 -2 or -__ = ___ 5 5 Left Side = Right Side The solution, m = -__6 , is correct. 5 x a NEL 8.1 Solving Equations: ax = b, __ __ a = b, x = b 295  c) Isolate the variable by applying the opposite operation. 1 k = -3__ -2__ 1 2 2 What is the sign of the -2__ 2 ( 1 = -3__ 1 k ÷ -2__ 2) 1 ÷ -2__ 1 2 ( 2) quotient when a negative is divided by a negative? k = -__ 2 5 ( ) 7 ÷ -__ 2 -7 -5 4 = ___ ÷ ___ -4 ÷ (-3) = __ 3 2 2 = -7 ___ -5 One way to divide = 7 or 1__ 2 __ fractions with the same 5 5 denominator is to simply divide the numerators. How else could you divide these fractions? Check: Left Side = -2__1k 1 Right Side = -3__ 2 2 = -2__1 1__ 2 5 2 ( ) 5 __ = -__ 2 5 7 ( ) 35 = -___ 10 = - 7 or -3__ __ 1 2 2 Left Side = Right Side The solution, k = __7 , is correct. 5 Show You Know Solve. a) 3x = -__ 2 3 x 5 b) __ = __ 2 6 1 3 c) -1__y = 1__ 4 4 296 Chapter 8 NEL  Example 2: Solve One-Step Equations With Decimals Solve each equation. Check each solution. a) -1.2x = -3.96 r b) _____ = -4.5 0.28 Solution a) Solve by applying the opposite operation. -1.2x = -3.96 -1.2x -3.96 ______ = ______ -1.2 -1.2 -4 ÷ -1 = 4 x = 3.3 C 3.96 + - ÷ 1.2 + - = 3.3 Check: Left Side = -1.2x Right Side = -3.96 = -1.2(3.3) = -3.96 C 1.2 + - × 3.3 = -3.96 Left Side = Right Side The solution, x = 3.3, is correct. b) Isolate the variable by applying the opposite operation. r = -4.5 _____ 0.28 0.28 × 4.5 ≈ 0.3 × 4 r = 0.28 × (-4.5) 0.28 × _____ ≈ 1.2 0.28 r = -1.26 What is the sign of C 0.28 × 4.5 + - = -1.26 the product when a positive is multiplied by a negative? Check: r Left Side = _____ Right Side = -4.5 0.28 -1.26 = ______ C 1.26 + - ÷ 0.28 = -4.5 0.28 = -4.5 Left Side = Right Side The solution, r = -1.26, is correct. Show You Know Solve and check. u a) ___ = 0.8 1.3 b) 5.5k = -3.41 x a NEL 8.1 Solving Equations: ax = b, __ __ a = b, x = b 297  a Example 3: Apply Equations of the Form ̲̲ x =b The formula for speed is s = __d, Did You Know? t where s is speed, d is distance, At Calgary Stampeders football and t is time. The length of a games, a white horse Canadian football field, including named Quick Six the end zones, is 137.2 m. If a charges the length of horse gallops at 13.4 m/s, how the field each time much time would it take the the Stampeders score a touchdown. The horse to gallop the length of the rider, Karyn Drake, field? Express your answer to the carries the team flag. nearest tenth of a second. Digital rights not available. Solution Substitute the known values into the formula. d s = __ t 137.2 13.4 = ______ t Why do you t × 13.4 = t × 137.2 ______ multiply by t? t t × 13.4 = 137.2 t × 13.4 = ______ ________ 137.2 13.4 13.4 130 ÷ 13 = 10 t ≈ 10.2 C 137.2 ÷ 13.4 = 10.23880597 The horse would take approximately 10.2 s to gallop the length of the field. Check: For a word problem, check your answer by verifying that the solution is consistent with the information given in the problem. Calculate the speed by dividing the distance, 137.2 m, by the time, 10.2 s. C 137.2 ÷ 10.2 = 13.45098039 Because the time was not exactly 10.2 s, this calculated speed of about 13.45 m/s is not exactly the same as the speed of 13.4 m/s given in the problem. But since these speed values are close, the answer is reasonable. Show You Know If a musher and her dog-team average 23.5 km/h during a dogsled race, how long will it take to sled 50 km? Express your answer to the nearest tenth of an hour. 298 Chapter 8 NEL  Example 4: Write and Solve Equations Winter Warehouse has winter jackets on sale at 25% off the regular price. If a jacket is on sale for $176.25, what is the regular price of the jacket? Solution Let p represent the regular price of the jacket. The sale price is 75% of the regular price. How do you know that So, the sale price is 0.75p. the sale price is 75% of the regular price? Since the sale price is $176.25, an equation that represents the situation is 0.75p = 176.25 0.75p _______ ______ = 176.25 0.75 0.75 200 ÷ 1 = 200 p = 235 C 176.25 ÷ 0.75 = 235. The regular price of the jacket is $235. Check: The price reduction is 25% of $235. 0.25 × $235 = $58.75 The sale price is $235 - $58.75. $235 - $58.75 = $176.25 The calculated sale price agrees with the value given in the problem, so the answer, $235, is correct. Show You Know Winter Warehouse is selling mitts at 30% off the regular price. If the sale price is $34.99, what is the regular price of the mitts? x a NEL 8.1 Solving Equations: ax = b, __ __ a = b, x = b 299  Key Ideas You can solve equations in various ways, including using diagrams using concrete materials g __ __ =2 2x = 0.10 2 3 + _2 = N A D A A C 3 10 CENTS 2009 0 +1 +2 g __ 2 CENT CENT = N A D A S S 5 5 A C 2009 2009 2009 10 CENTS + _2 3 x = 0.05 0 +1 +2 g __ g __ 2 2 g 4 1 g = __ or 1__ 3 3 using an algebraic method -1.4 _____ p = -0.8 -1.4 ( ) p × _____ = p × (-0.8) p -1.4 = p × (-0.8) p × (-0.8) -1.4 _________ _____ = -0.8 -0.8 1.75 = p You can check solutions by using substitution. -1.4 Left Side = _____ p Right Side = -0.8 -1.4 C 1.4 + - = _____ ÷ 1.75 = -0.8 1.75 = -0.8 Left Side = Right Side The solution, p = 1.75, is correct. To check the solution to a word problem, verify that the solution is consistent with the facts given in the problem. 300 Chapter 8 NEL Check Your Understanding Communicate the Ideas y 5 1. To solve the equation __ = __, John first multiplied both sides by 3. 2 3 a) Do you think that John’s first step is the best way to isolate the variable y? Explain. b) How would you solve the equation? 2. When Ming solved 0.3g = 0.8, her value for g was 2.666…. She expressed this to the nearest tenth, or 2.7. When Ming checked by substitution, she found that the left side and the right side did not exactly agree. Left Side = 0.3g Right Side = 0.8 = 0.3(2.7) = 0.81 a) How could Ming make the left side and right side agree more closely? b) Did Ming’s check show that the solution was correct? Explain. 3. The length of Shamika’s stride is 0.75 m. Both Amalia and Gustav were trying to calculate how many strides it would take Shamika to walk 30 m from her home to the bus stop. a) Amalia represented the situation with the equation 0.75p = 30. Explain her thinking. 30 b) Gustav represented the situation with the equation ___ = 0.75. Explain p his thinking. c) Whose equation would you prefer to use? Explain. Practise 4. Write an equation that is represented by 7. Solve. the model shown. Then, solve it. 3 x 6 a) __ = __ b) 2y = -__ 5 4 5 7 4 2 1 2009 CA A 2 00 NADA = c) -__ = -__ n d) 2__ w = 1__ 25 AN D 9 Cents A C 2009 6 3 3 6 CA NADA For help with #8 and #9, refer to Example 2 on page 297. For help with #5 to #7, refer to Example 1 on 8. Solve and check. pages 294–296. e 3 a) -5.6x = 3.5 b) _____ = -0.75 5. Model the solution to the equation 4x = __ -2.2 using a number line. 4 9. Solve. 6. Solve. h a) ___ = 3.6 b) 1.472 = 0.46c -5 a) 2v = ___ x 2 b) __ = __ 4.1 6 2 5 4 1 c) __ = -1__ a 1 1 d) -1__ x = -2__ 3 4 2 4 x a NEL 8.1 Solving Equations: ax = b, __ __ a = b, x = b 301 For help with #10 and #11, refer to Example 3 on 16. The diameter, d, of a circle is related to the page 298. C = π. circumference, C, by the formula __ 10. Solve and check. d 1.1 4.8 Calculate the diameter of a circle with a a) -5.5 = ___ b) ___ = 6.4 circumference of 54.5 cm. Express your a m answer to the nearest tenth of a centimetre. 11. Solve. Express each solution to the nearest hundredth. 17. A regular polygon has a perimeter of 2.02 -4.3 34.08 cm and a side length of 5.68 cm. a) _____ = 0.71 b) -7.8 = _____ Identify the polygon. n x Apply Literacy Link 108° 12. The average speed of a vehicle, s, A regular polygon has equal 108° 108° is represented by the formula sides and equal angles. For d where d is the distance example, a regular pentagon s = __ has five equal sides. Each t 108° 108° angle measures 108°. driven and t is the time. a) If Pablo drove at an average speed of 85 km/h for 3.75 h, what 18. One year, a student council sold 856 copies distance did he drive? of a school yearbook. Four fifths of the b) If Sheila drove 152 km at an average students at the school bought a copy. How speed of 95 km/h, how much time many students did not buy a yearbook? did her trip take? 19. A score of 17 on a math test results in a 13. A roll of nickels is worth $2.00. Write and mark of 68%. What score would give solve an equation to determine the number you a mark of 100%? of nickels in a roll. 3 20. The area of Nunavut is about 4__ times 8 the area of the Yukon Territory. Nunavut covers 21% of Canada’s area. What percent of Canada’s area does the Yukon Territory cover? 14. Write and solve an equation to determine the side length, s, of a square with a perimeter of 25.8 cm. 15. Without solving the equation - 5 __ = -1.3, d Lewis Lake, Yukon predict whether d is greater than or less than 0. Explain your reasoning. 302 Chapter 8 NEL 21. Dianne spends 40% of her net income on Extend rent and 15% of her net income on food. If 26. Solve. she spends a total of $1375 per month on rent and food, what is her net monthly 1 1 5 a) __ + __ = __ x 0.45 0.81 b) -_____ = -_____ 3 6 6 1.8 z income? y y 1 f c) __ - __ = -___ d) _____ = 2.6 - 3.5 22. Ellen and Li play on the same basketball 4 3 10 0.55 team. In one game, Ellen scored one tenth of the team’s points and Li scored one fifth of 27. Solve. Express each solution to the the team’s points. Together, Ellen and Li nearest hundredth. scored 33 points. How many points did the a) 0.75 + 1.23 = -3.9t team score altogether? 6.3 b) ___ = 2(-4.05) h 23. Sailaway Travel has a last-minute sale on a Caribbean cruise at 20% off. Their 28. Solve and check. advertisement reads, “You save $249.99.” What is the sale price of the cruise? 1 3 a) x ÷ __ = -__ 2 4 2 ( 3)1 b) t ÷ -__ = -__ 2 c) 5 __ ÷y= 2 __ 2 3 d) -__ ÷ g = ___ 24. Organizers of the Canadian Francophone 6 3 5 10 Games hope to attract 500 volunteers to help host the games. The organizers predict 29. a) A jar contains equal numbers of nickels that there will be about three and half times and dimes. The total value of the coins is as many experienced volunteers as first-time $4.05. How many coins are in the jar? volunteers. About how many first-time b) A jar contains a mixture of nickels and volunteers do they expect to attract? dimes worth a total of $4.75. There are three times as many nickels as dimes. 25. A square piece of paper is folded in half to How many dimes are there? make a rectangle. The perimeter of the rectangle is 24.9 cm. What is the side length 30. A cyclist is travelling six times as fast as a of the square piece of paper? pedestrian. The difference in their speeds is 17.5 km/h. What is the cyclist’s speed? Solve parts a) and b) in at least two different ways. Write and solve an equation as one of the methods for each part. Share your solutions with your classmates. Three dried figs contain about 1.2 mg of iron. a) What is the mass of iron in one dried fig? b) Teenagers need about 12 mg of iron per day. How many dried figs would you have to eat to get your recommended daily amount of iron? c) Write a formula that relates the mass of iron to the number of figs. Use your equation to calculate the mass of iron in eight figs. d) Use your formula in part c) to determine the number of figs that contain 1.8 mg of iron. x a NEL 8.1 Solving Equations: ax = b, __ __ a = b, x = b 303  8.2 Solving Equations: x ̲ ax + b = c, a + b = c Focus on… After this lesson, you will be able to… Two of Canada’s model problems highest measured with linear equations waterfalls are in British involving two Columbia. Takakkaw operations Falls, is in Yoho solve linear National Park, 27 km equations with west of Lake Louise. Its rational numbers height is 254 m. This is using two operations 34 m more than half the height of Della Falls in Strathcona Park on Web Link Vancouver Island. To learn more about Choose a variable to waterfalls in Canada and around the world, go to represent the height of www.mathlinks9.ca and Della Falls. Then, write follow the links. and solve an equation to find the height of Della Falls. Della Falls, Vancouver Island Materials Explore Equations With Two Operations coins cups or small containers 1. a) How does the diagram model the solution to the equation paper clips 2x + 0.30 = 0.50? b) What is the solution? A 2 00 A 2 00 25 25 AN D 9 AN D 9 Cents A Cents A + = C C CENT A 2 00 25 AN D 9 S 5 Cents A 2009 C A 2 00 A 2 00 25 25 AN D 9 AN D 9 Cents A Cents A N A D N A D + = A A C C A A C C 2009 2009 10 CENTS 10 CENTS CENT A 2 00 25 AN D 9 S 5 Cents A 2009 C 304 Chapter 8 NEL  2. a) Explain how the second part of the diagram in #1 can model the equation 0.10y + 0.30 = 0.50. What is the solution? Explain. b) How does the second part of the diagram in #1 also model the z 3 = __ 1 ? What is the solution? solution to the equation ___ + ___ 10 10 2 3. Describe how you would use manipulatives or diagrams to model the solution to each of the following. a) 3x + 0.05 = 0.26 b) 0.01x + 0.05 = 0.08 x 1 7 c) __ + __ = ___ 4 5 10 4. Work with a partner to explore how to model the solution to the equation 2x - 0.11 = 0.15. Share your models with other classmates. Reflect and Check 5. a) How can you model solutions to equations of the form ax + b = c x + b = c using manipulatives or diagrams? and __ a b) Think of other ways to model the solutions. Explain how you would use them. 6. The tallest waterfall in the world is Angel Falls in Did You Know? Venezuela, with a height of Canada’s tallest free- about 0.8 km. This height standing structure is is 0.08 km less than twice the CN Tower, with a height of about 550 m. the height of Della Falls. This is about 250 m Write and solve an less than the height equation to determine the of Angel Falls. height of Della Falls in kilometres. Check that your answer agrees with the height in metres you determined at the beginning of this section. Angel Falls, Venezuela x NEL 8.2 Solving Equations: ax + b = c, __ a+b=c 305  Link the Ideas Example 1: Solve Two-Step Equations With Fractions Solve and check. 1 3 a) 2x + ___ = __ k 1 3 b) __ - __ = -1__ 10 5 3 2 4 Solution To isolate the variable in a a) 2x + ___ 3 1 = __ two-step equation, use the reverse 10 5 order of operations. Add or subtract first, and then multiply 1 - ___ 2x + ___ 3 - ___ 1 = __ 1 or divide. 10 10 5 10 3 - ___ 2x = __ 1 5 10 6 - ___ 2x = ___ 1 10 10 5 2x = ___ 10 1 __ ÷ 2 = __ × __ 1÷2 __ 1 1 2x ÷ 2 = __ 2 1 2 2 2 1 x = __ 4 3 on a number line. 1 = __ Check by modelling the equation 2x + ___ 10 5 + _3 5 This diagram models 0 +1 the original equation 1 3 2x 1 __ 2x + ___ = __. How does 10 5 10 5 it show that 2x = ___? 10 Now show the value of x. + _3 5 0 +1 2x 1 __ 10 x x 5 or __ The second diagram shows that x = ___ 1. 20 4 1 , is correct. The solution, x = __ 4 306 Chapter 8 NEL  k 1 3 b) __ - __ = -1__ 3 2 4 k - __ __ 7 1 = -__ 3 2 4 You may prefer to work with integers than to perform fraction operations. Change from fractions to integers by multiplying by a common multiple of the denominators. 12 × __ 3 2 7 1 = 12 × -__ k - 12 × __ 4 ( ) A common multiple of 4k - 6 = -21 the denominators 3, 2, 4k - 6 + 6 = -21 + 6 and 4 is 12. 4k = -15 4k = ____ ___ -15 4 4 15 k = -___ 4 Check: 1 k - __ Left Side = __ 3 Right Side = -1__ 3 2 4 15 ÷ 3 - __ = -___ 1 4 2 1 - __ -15 × __ = ____ 1 4 3 2 1 -5 - __ = ___ 4 2 2 -5 - __ = ___ 4 4 -7 or -1__ = ___ 3 4 4 Left Side = Right Side 15 , is correct. The solution, k = -___ 4 Show You Know Solve and check. 1 3 a) 2y + __ = __ 2 4 n 3 3 b) __ - __ = 2__ 2 4 8 x NEL 8.2 Solving Equations: ax + b = c, __ a+b=c 307  Example 2: Solve Two-Step Equations With Decimals a - 2.5 = -3.7 and check the solution. Solve ___ 2.8 Solution a - 2.5 = -3.7 ___ 2.8 a - 2.5 + 2.5 = -3.7 + 2.5 ___ 2.8 a = -1.2 ___ 2.8 a = 2.8 × (-1.2) 2.8 × ___ 2.8 2.8 × (-1.2) ≈ 3 × (-1) ≈ -3 a = -3.36 Check: a - 2.5 Left Side = ___ Right Side = -3.7 2.8 -3.36 - 2.5 = ______ 2.8 = -1.2 - 2.5 = -3.7 Left Side = Right Side The solution, a = -3.36, is correct. Show You Know h + 3.3 = 1.8 and check the solution. Solve ___ 1.6 Example 3: Apply Two-Step Equations With Decimals Colin has a long-distance telephone plan that charges 5¢/min for long-distance calls within Canada. There is also a monthly fee of $4.95. One month, Colin’s total long-distance charges were $18.75. How many minutes of long-distance calls did Colin make that month? Solution Let m represent the unknown number of minutes. The cost per minute is 5¢ or $0.05. The cost of the phone calls, in dollars, is 0.05m. The total cost for the month is the cost of the calls plus the monthly fee, or 0.05m + 4.95. The total cost for the month is $18.75. 308 Chapter 8 NEL  An equation that represents the situation is 0.05m + 4.95 = 18.75. 0.05m + 4.95 - 4.95 = 18.75 - 4.95 0.05m = 13.80 13.80 ____ _____ 15 0.05m 13.80 ______ = ______ 0.05 ≈ 0.05 0.05 0.05 ≈ 300 m = 276 Colin made 276 min of long-distance calls that month. Check: The cost for 276 min at 5¢/min is $0.05 × 276. 0.05 × 276 = 13.80 The total cost for the month is $13.80 + $4.95, which equals $18.75. This total cost agrees with the value stated in the problem. Show You Know Suppose that Colin changes to a cheaper long-distance plan. This plan charges 4¢/min for long-distance calls within Canada, plus a monthly fee of $3.95. For how many minutes could he call long distance in a month for the same total long-distance charge of $18.75? Key Ideas You can determine or check some solutions by using a model. 1 __ 3u + __ =7 + _7 8 8 8 0 +1 3u _1 8 6 + _7 3u = __ 8 8 0 +1 3u _1 8 u u u 2 __ u = __ or 1 8 4 To isolate the variable in a two-step equation, use the reverse order of operations. Add or subtract first, and then multiply or divide. 0.4w - 1.5 = 0.3 0.4w - 1.5 + 1.5 = 0.3 + 1.5 0.4w = 1.8 0.4w ___ _____ 1.8 = 0.4 0.4 w = 4.5 x NEL 8.2 Solving Equations: ax + b = c, __ a+b=c 309  To solve two-step equations involving fractions, you may prefer to rewrite the equation and work with integers than to perform fraction operations. w __ __ 3 1 - = ___ 5 2 10 To work with integers, multiply all terms by a common multiple of the denominators. For the denominators 5, 2, and 10, a common multiple is 10. w 3 1 10 × __ - 10 × __ = 10 × ___ 5 2 10 2w - 15 = 1 2w - 15 + 15 = 1 + 15 2w = 16 16 2w = ___ ___ 2 2 w=8 You can check solutions by using substitution. w __ 3 1 Left Side = __ - Right Side = ___ 5 2 10 8 3 = __ - __ 5 2 16 ___ ___ 15 = - 10 10 1 = ___ 10 Left Side = Right Side The solution, w = 8, is correct. To check the solution to a word problem, verify that the solution is consistent with the facts given in the problem. Check Your Understanding Communicate the Ideas 1. Explain how the diagrams model the equation x + __ __ 5 and its solution. What is the solution? 1 = __ 2 4 8 + _5 8 0 +1 _x _1 2 4 + _5 8 0 +1 _x _x 2 2 x 310 Chapter 8 NEL  2. Ryan solved 2r + 0.3 = 0.7 as follows. Do you agree with his solution? Explain. 2r __ 0.7 + 0.3 = ___ 2 2 r + 0.3 = 0.35 r + 0.3 - 0.3 = 0.35 - 0.3 r = 0.05 3. Jenna did not want to perform fraction operations to solve the x - __ equation __ 5 , so she first multiplied both sides by 54. Is this 1 = __ 2 9 6 the common multiple you would have chosen? Explain. 4. a) Milos solved 0.05x - 0.12 = 0.08 by multiplying both sides by 100 and then solving 5x - 12 = 8. Show how he used this method to determine the correct solution. x b) When Milos was asked to solve _____ - 0.12 = 0.08, he 0.05 reasoned that he could determine the correct solution by x - 12 = 8. Do you agree with his reasoning? Explain. solving __ 5 Practise 5. Write an equation that is modelled by For help with #9 and #10, refer to Example 2 on the following. Then, solve it. page 308. A N A D A 2 00 25 9. Solve and check. AN D 9 Cents A C A 2009 + 10 CENTS = C CENT x a) ___ + 2.5 = -1 S 5 CENT A 2 00 25 2009 0.6 AN D S 5 9 Cents A 2009 C r b) 0.38 = 6.2 - ___ 1.2 6. Model the equation 3x + 0.14 = 0.50 using concrete materials. Solve using your model. 10. Solve. n a) -0.02 - ___ = -0.01 For help with #7 and #8, refer to Example 1 on 3.7 pages 306–307. k b) ______ + 0.67 = 3.47 7. Solve. -0.54 2 3 a) 4y - __ = __ 1 5 b) 2d - __ = __ For help with #11 and #12, refer to Example 3 on 5 5 2 4 pages 308–309. n 2 1 c) __ + 1__ = __ 4 1 3 d) __ - 2__ r = ___ 11. Solve and check. 2 3 6 5 2 10 a) 2 + 12.5v = 0.55 8. Solve. b) -0.77 = -0.1x - 0.45 1 2 a) 1__ = 4h + __ 4 3 1 b) __ x + __ = __ 2 3 3 4 2 12. Solve. 3 d 3 2 1 7 a) 0.074d - 3.4 = 0.707 c) __ - __ = __ d) -4__ = -3__ + ___ g 4 3 8 5 5 10 b) 67 = 5.51 + 4.3a x NEL 8.2 Solving Equations: ax + b = c, __ a+b=c 311 Apply 19. The greatest average annual snowfall in Alberta is on the Columbia Icefield. The 13. The cost of a pizza is $8.50 plus $1.35 per greatest average annual snowfall in Manitoba topping. How many toppings are on a pizza is at Island Lake. An average of 642.9 mm of that costs $13.90? snow falls on the Columbia Icefield in a year. 14. Hiroshi paid $34.95 to rent a car for a day, This amount of snow is 22.5 mm less than plus 12¢ for each kilometre he drove. The twice the annual average at Island Lake. What total rental cost, before taxes, was $55.11. is the average annual snowfall at Island Lake? How far did Hiroshi drive that day? 15. On Saturday morning, Marc had a quarter of his weekly allowance left. He spent a total of $6.50 on bus fares and a freshly squeezed orange juice on Saturday afternoon. He then had $2.25 left. How much is his weekly allowance? 16. Nadia has a summer job in an electronics store. She is paid $400 per week, plus 5% commission on the total value of her sales. a) One week, when the store was not busy, Nadia earned only $510.30. What was Columbia Icefield the total value of her sales that week? b) Nadia’s average earnings are $780 per 20. During a camping trip, Nina was making a week. What is the average value of her lean-to for sleeping. She cut a 2.5-m long weekly sales? post into two pieces, so that one piece was 26 cm longer than the other. What was the length of each piece? Digital rights not available. 21. The average monthly rainfall in Victoria in July is 2.6 mm less than one fifth of the amount of rain that falls in Edmonton in the same period. Victoria averages 17.6 mm of rainfall in July. What is the average monthly 17. Benoit was helping his family build a new rainfall in Edmonton in July? fence along one side of their yard. The total length of the fence is 28 m. They worked for 22. The temperature in Winnipeg was 7 °C and two days and completed an equal length of was falling by 2.5 °C/h. How many hours did fence on each day. On the third day, they it take for the temperature to reach -5.5 °C? completed the remaining 4.8 m of fence. What length of fence did they build on 23. Max and Sharifa are both saving to buy the each of the first two days? same model of DVD player, which costs $99, including tax. Max already has $31.00 18. The perimeter of a regular hexagon is and decides to save $8.50 per week from 3.04 cm less than the perimeter of a regular now on. Sharifa already has $25.50 and pentagon. The perimeter of the regular decides to save $10.50 per week from now hexagon is 21.06 cm. What is the side on. Who can pay for the DVD player first? length of the regular pentagon? Explain. 312 Chapter 8 NEL 24. A cylindrical storage tank that holds 375 L 29. Solve. Express each solution to the of water is completely full. A pump removes nearest hundredth. water at a rate of 0.6 L/s. For how many a) 0.75 + 0.16y + 0.2y = 0.34 minutes must the pump work until 240 L of -1.85 water remain in the tank? b) ______ = 2.22 - 0.57s 0.74 25. The average distance of Mercury from the 30. Solve. sun is 57.9 million kilometres. This distance 0.2 a) ___ + 0.8 = 1.2 is 3.8 million kilometres more than half the x average distance of Venus from the sun. 1 4 1 What is the average distance of Venus from b) __ - __ = -__ 2 n 4 the sun? 3.52 c) -_____ - 1.31 = 1.19 x h 26. Create an equation of the form __ + b = c a 5 1 3 d) 4__ = 3__ - __ with each given solution. Compare your 6 3 y equations with your classmates’ equations. 2 a) __ 31. Determine the 3 value of x. b) -0.8 A = 1.69 cm2 27. Write a word problem that can be solved using an equation of the form ax + b = c. Include at least one decimal or fraction. (3 + 2x) cm Have a classmate solve your problem. 32. A freight train passes through a 750-m long Extend tunnel at 50 km/h. The back of the train exits the tunnel 1.5 min after the front of the 28. Solve. train enters it. What is the length of the 3 w 5 1 a) __ + __ = __ - __ train, in metres? 2 4 6 2 3 2 ( 9) 1 1 b) __ -__ = 4__x + __ 4 2 3 Revelstoke, British Columbia A slice of canned corned beef contains about 0.21 g of sodium. This much sodium is 0.01 g more than the mass of sodium in four slices of roast beef. What is the mass of sodium in a slice of roast beef? a) Write an equation that models the situation. b) Solve the equation in two different ways. c) Which of your solution methods do you prefer? Explain. x NEL 8.2 Solving Equations: ax + b = c, __ a+b=c 313  8.3 Solving Equations: a(x + b) = c Focus on… After this lesson, you will be able to… model problems with linear equations that include grouping symbols on one side solve linear equations that include grouping symbols on one side Each year, Canada’s Prairie Provinces produce tens of millions of tonnes of Did You Know? grains, such as wheat, barley, and canola. The growth of a grain crop partly Farms account for depends on the quantity of heat it receives. One indicator of the quantity of only about 7% of the heat that a crop receives in a day is the daily average temperature. This is land in Canada. About 80% of Canada’s defined as the average of the high and low temperatures in a day. farmland is located in How can you calculate the daily average temperature on a day when the the Prairie Provinces: Alberta, Saskatchewan, high temperature is 23 °C and the low temperature is 13 °C? If the low and Manitoba. temperature is 10 °C, how could you determine the high temperature that would result in a daily average temperature of 15 °C? What equations can you use to represent these situations? Materials Explore Equations With Grouping Symbols coins paper cups or small 1. Explain how the diagram models A 2 00 25 AN D 9 Cents A containers the solution to the equation A N A D A N A D = C A A C C 2009 2009 10 CENTS 10 CENTS CENT paper clips 2(x + 0.10) = 0.30. What is the S 5 solution? 2009 A 2 00 CENT CENT 25 AN D 9 Cents A S S 5 5 = C 2009 2009 A