CHAPTER 10 Solving Linear Equations GET READY 524 Math Link 526 10.1 Warm Up 527 10.1 Modelling and Solving One-Step Equations: a b, a b 528 10.2 Warm Up 537 10.2 Modelling and Solving Two-Step Equations: a + b c 538 10.3 Warm Up 548 10.3 Modelling and Solving Two-Step Equations: a + b c 549 10.4 Warm Up 557 10.4 Modelling and Solving Two-Step Equations: a( + b) c 558 Chapter Review 568 Practice Test 574 Wrap It Up! 578 Key Word Builder 579 Math Games 580 Challenge in Real Life 581 Answers 582
Substituting Values into Equations To substitute values into an equation: Step 1: Replace the variable. A letter that represents an unknown number Step 2: Follow the order of operations: brackets first multiply and/or divide in order from left to right add and/or subtract in order from left to right Find the value of y when 5. y 2( 4) + 3 Substitute. y 2(5 4) + 3 Brackets first. y 2(1) + 3 Multiply. y 2 + 3 Add. y 5 1. Find the value of y when 4. a) y 2( 1) + 6 y 2(4 1) + 6 y 2( ) + 6 y + 6 y b) y (3 + 7) 4 y (3 + 7) 4 y ( 7) 4 y 4 y Modelling and Solving One-Step Equations To solve a word problem, change the words into symbols, letters, and numbers to make an equation. Rewrite as A number increased by 5 is 8. 4 times a number is 24. 4 24 n + 5 8 4 24 To solve an equation, use the opposite operation to get alone on 1 side of the equation. opposite operation an operation that undoes another operation and + are opposite operations, and are opposite operations Eamples: n + 5 8 n + 5 5 8 5 n + 5 5 8 5 n 3 What you do to 1 side, do to the other. 4 24 4 24 4 4 6 524 MHR Chapter 10: Solving Linear Equations
2. Write each sentence as an equation. a) Seven more than a number is twelve. n + b) Three less than a number is eleven. c) Four times a number is twenty-eight. d) When a number is divided by nine, the result is nine. Solving Two-Step Equations To solve a 2-step equation, find the value of. 2 + 3 7 2 + 3 3 7 3 Subtract to undo addition. 2 + 3 3 7 3 2 4 Divide to undo multiplication. 2 4 2 2 2 Get alone on 1 side of the equation. 3. For each equation, circle the first operation you should do to both sides. a) 2n + 4 18 Add 4 or subtract 4. c) 8y 70 94 Add 70 or subtract 70. b) 3 + 5 17 Add 5 or subtract 5. d) 27 7q + 6 Add 6 or subtract 6. 4. Solve the equations. a) 5j + 9 29 b) 2t 2 14 5j + 9 29 5j 5 j j Get Ready MHR 525
Modelling Equations change a word into a code You can use equations to encrypt a password. equation 2 epressions joined with an equal sign eamples: 5 8 or 2y + 3 15 Jim s password is weather. He gives each letter in the alphabet a number value. 1 a 2 b 3 c 4 d 5 e 6 f 7 g 8 h 9 i 10 j 11 k 12 l 13 m 14 n 15 o 16 p 17 q 18 r 19 s 20 t 21 u 22 v 23 w 24 25 y 26 z The number sequence for the password weather is: w e a t h e r 23 5 1 20 8 5 18 Jim uses the equation y 3 + 2 to encrypt the password. This gives each letter in the password a different number value. Find the new values of the letters. w 23 e 5 a 1 y 3(23) + 2 y 3( ) + 2 y 69 + 2 y + 2 y y t h r Jim s code for weather is 71. 526 MHR Chapter 10: Solving Linear Equations
10.1 Warm Up 1. Write an equation for each diagram. represents represents represents 1 a) b) c) d) 2. Show each equation using algebra tiles. a) 3 + 5 1 b) 2 + 1 5 3. Multiply or divide. a) 6 ( 5) b) 3 8 + - + - + + - + - - + - c) ( 4) ( 7) d) ( 24) 6 e) 15 3 f) ( 12) ( 4) 4. Fill in the boes. a) + 4 7 b) + 8 10 c) 5 6 d) 1 4 e) 2 12 f) 3 15 g) 4 10 h) 6 3 10.1 Warm Up MHR 527
10.1 Modelling and Solving One-Step Equations: a b, a b linear equation an equation whose points on a graph lie along a straight line eamples: y 4, d 2 c, 5w + 1 t numerical coefficient a number that multiplies the variable eample: 4k + 3 numerical coefficient variable a letter used to represent an unknown number eample: 2y + 3 variable constant a number that is not connected to a variable eample: 2 + 5 3 constant Working Eample 1: Solve an Equation Solve each equation. a) 3 12 Solution Method 1: Solve by Inspection 3 12 3 12 3 4 12 So, What number times 3 equals 12? represents a positive variable. represents a negative variable. represents. represents 1. Method 2: Solve Using Models and Diagrams Use algebra tiles. 3 black tiles 3 12 white tiles 12 3 black tiles 12 white tiles How many white tiles 1 black tile? So,. 528 MHR Chapter 10: Solving Linear Equations
r b) 7 2 Solution Method 1: Solve by Inspection r 7 2 2 14 2 7 7 What number divided by 2 equals 7? Method 2: Solve Using Models and Diagrams 1 whole circle r Divide the circle into 2 parts. Shade half the circle. -r 2 r 2 Add white tiles to show 7. So,. -r 2 r 2 2 7 r Half the circle shows 2, so multiply the amount of shaded circle by 2 to represent r. Then, multiply the number of white squares by 2 so that the equation is balanced. -r white squares represent 14. So, r. To make r positive, multiply both sides of the equation by 1. 1( r) 1( 14) So, r. 10.1 Modelling and Solving One-Step Equations: a b, a b MHR 529
Solve each equation. Use inspection, models, or diagrams. Check your work. Remember, + + + - - + + - - - + - a) 4 8 b) y 5 2 4 4 8 y 2 5 2 530 MHR Chapter 10: Solving Linear Equations
Working Eample 2: Divide to Apply the Opposite Operation Simone uses a spring to do an eperiment. The equation F 12d shows how much force the spring has. F force in newtons, N d distance the spring stretches, in cm Simone pushes the spring together with a force of 84 N. What distance is the spring compressed? pressed together Solution F 12d Substitute 84 for F. 84 12d The opposite of 12 is 12. 84 12d C 84 12? d The spring is compressed cm. The spring was pushed in, so d is negative. 84 12d 12 ( 7) Solve by applying the opposite operation. Check your answer. a) 5b 45 5b 45 b) 6f 12 b 5b 45 5 10.1 Modelling and Solving One-Step Equations: a b, a b MHR 531
Working Eample 3: Multiply to Apply the Opposite Operation In January, the average afternoon temperature in Edmonton is 1 3 afternoon temperature in Yellowknife. The average afternoon temperature in Edmonton is 8 C. What is the average afternoon temperature in Yellowknife? the average Solution Let t represent the average afternoon temperature in Yellowknife. 1 3 the average afternoon temperature in Yellowknife 1 t t or 3 3 t The equation to model the problem is 8. 3 t 3 t 3 t 8 The opposite of 3 is 3. 3 8 3 The average afternoon temperature in Yellowknife is C. t 3 24 3 8 Solve by applying the opposite operation. Check your answer. d 3 5 d 3 5 d d 5 3 5 532 MHR Chapter 10: Solving Linear Equations
1. Five times a number is 15. a) Write an equation for this sentence. b) Draw a picture to show how to find the unknown number. 2. Raj is solving the equation 9 n 4. n 9 n 9 4 X ( 9) -4 X ( 9) n 36 Raj s solution is wrong. Eplain where he made his mistake. 3. Write the equation modelled by each diagram. a) t t t represents represents 1 b) -w 2 t c) m 3 d) -n -n m 3 10.1 Modelling and Solving One-Step Equations: a b, a b MHR 533
4. Solve by inspection. a) 2j 12 b) 5n 25 2 12 5 25 j n 5. Solve each equation using the opposite operation. Check your answers. a) 4s 12 4s 12 b) 36 3j s c) t 3 8 d) 9 10 3 t 8 9 10 t 534 MHR Chapter 10: Solving Linear Equations
6. Nakasuk s snowmobile can travel for 13 km on 1 L of gas. a) How far can he travel on 2 L of gas? b) How far can he travel on 3 L of gas? c) How far can he travel on 10 L of gas? d) Nakasuk visits his aunt who lives 312 km away. How many litres of gas will he need? Use the equation 13 312 to help you. 7. Lucy is making 2 pairs of mittens. She has 72 cm of trim to sew around the cuffs. How much trim does she have for each mitten? a) How many mittens in total is she making? b) Let t represent the amount of trim for 1 mitten. Equation to find the amount of trim for 1 mitten: t c) Solve the equation to find the amount of trim for 1 mitten. Sentence: 10.1 Modelling and Solving One-Step Equations: a b, a b MHR 535
Have you ever dropped a ball of Silly Putty on a hard surface? It bounces! The greater the height from which it is dropped, the higher it bounces. With a partner, perform an eperiment with Silly Putty. a) Tape a metre stick vertically ( ) to a wall or desk. The 0-cm end should touch the floor. One person sits on the floor looking at the ruler. The other person stands net to the ruler, holding the putty. b) Choose 4 different heights from which to drop the putty. Record the heights in the table. c) Drop the ball of putty from the 4 different heights you chose in part b). Each time, the person on the floor watches to see the height the putty bounces. The person on the floor records the height of the bounce in the table. Round your answer to the nearest centimetre. d) Complete the chart to find k. Round your answers to 2 decimal places. Trial Height of Drop, h Height of First Bounce, b k b h Eample 20 cm 3 cm k 3 20 0.15 1 2 3 4 e) What do you notice about the numbers you recorded in the last column? f) Write an equation that shows the results of your eperiment. Substitute your value for k into the equation. b h 536 MHR Chapter 10: Solving Linear Equations
10.2 Warm Up 1. Write the equation for each diagram. a) b) 2. Evaluate. a) b + 6, when b 5 b) s 3, when s 12 b + 6 + 6 3. Find the missing number by inspection. a) 5 15 b) 4 24 c) 2 8 d) 7 3 3 7 4. Multiply. 3(2) 3 2 6 a) 10( 4) b) 6( 1) 5. Calculate. Use the number line to help you. 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 a) ( 5) + (+8) b) (1) + ( 4) c) ( 6) + ( 7) d) ( 3) + (2) e) (+8) ( 6) f) ( 5) ( 9) (+8) + (+6) Add the opposite ( 5) + ( ) g) ( 9) (+3) h) (+2) (2) 10.2 Warm Up MHR 537
10.2 Modelling and Solving Two-Step Equations: a + b c Working Eample 1: Modelling With a Balance Scale Solve the equation 23 4k + 3. Solution You can use a balance scale to model this equation. The numerical coefficient is, so draw 4 black tiles. k k k k the number that multiplies the variable Isolate the variable by removing To keep things balanced, remove grey tiles from the right. grey tiles from the left. k k k k There are There are k blocks on the right side of the scale. grey blocks on the left side of the scale. To balance the scale, each k block must have a mass of So, k. grey blocks. 23 4k + 3 4(5) + 3 20 + 3 538 MHR Chapter 10: Solving Linear Equations
Use the balance scale to solve each equation. Show your steps, then check your answer. a) 6n + 6 18 b) 13 9 + 2p So, n. 6n + 6 6( ) + 6 + 6 18 13 9 + 2p 9 + 2( ) 9 + 10.2 Modelling and Solving Two-Step Equations: a + b c MHR 539
Working Eample 2: Model With Algebra Tiles A cow sleeps 7 h a day. This is 1 h less than twice the amount an elephant sleeps in a day. How long does an elephant sleep? a) Write an equation for this situation. Solution unknown value number of hours an elephant sleeps Let e represent the number of hours an elephant sleeps. Twice what an elephant sleeps 2e 1 h less than twice what an elephant sleeps 2e A cow sleeps 7 h, so the equation is 2e 1 b) Solve the equation using algebra tiles. Solution The numerical coefficient is, so draw black tiles. e e the number that multiplies the variable To isolate the black tiles on 1 side of the equal sign, add both sides. grey tile to e e The negative 1-tile and positive 1-tile on the left side equal 0. black tile is equal to grey tiles. e e e 4 So, an elephant sleeps h a day. 540 MHR Chapter 10: Solving Linear Equations
Solve each equation using algebra tiles. a) 2 + 4 6 Fill in the missing tiles so your model represents the equation. Isolate the black tiles. Split up the tiles so you can see that 1 black tile is equal to white tiles. b) 3r 2 13 r 10.2 Modelling and Solving Two-Step Equations: a + b c MHR 541
Working Eample 3: Apply the Opposite Operations Solve 4w + 3 19 by applying the opposite operation. Solution To isolate a variable, follow the reverse order of operations. 4w + 3 19 4w + 3 3 19 3 Subtract 3 from both sides of the equation. 4w Divide both sides by 4. + and and 4 4 4 w Check using algebra tiles: Draw algebra tiles to represent the equation. Isolate the black tiles. Subtract grey 1-tiles from both sides. The 4 black tiles must have the same value as the 16 grey 1-tiles. Each black tile must have a value of w grey 1-tiles. w w w w w w w w w w w w Solve by applying the opposite operation. a) 3 7 5 3 7 + 5 + 3 b) 4 + 2g 12 4 + 2g 12 ( ) 2g 12 + ( ) 3 2g g 542 MHR Chapter 10: Solving Linear Equations
1. a) Draw algebra tiles to model 3 5 16. Isolate means to get the variable alone. b) To isolate, add tiles to both sides of the equation. Give 1 reason why you need to add this number of tiles. Hint: Use zero pairs. c) To solve for, divide both sides of the equation by. Give 1 reason why you need to divide by this number. 2. Solve each equation modelled by the algebra tiles. a) b) t t t 10.2 Modelling and Solving Two-Step Equations: a + b c MHR 543
3. Solve the equation modelled by each balance scale. Check your solution. a) 3 + 8 11 b) 13 4h + 5 Remove 8 grey tiles from each side. h 544 MHR Chapter 10: Solving Linear Equations
4. Complete the table. Equation First Operation to Solve Second Operation to Solve 4r 2 14 Add to each side. Divide both sides by. 22 10 + 2n 53 9k 1 3 3 9 5. Solve each equation and check your answer. a) 6r + 6 18 b) 4m 2 14 6r + 6 18 6r 6r r 6r + 6 18 6( ) + 6 + 6 10.2 Modelling and Solving Two-Step Equations: a + b c MHR 545
6. You buy lunch at Sandwich Epress. A sandwich costs $4. Each etra topping costs $2. You have $10. Use the equation 2e + 4 10 to find how many etra toppings you can get if you spend all of your money. MENU Your choice of etras, only $2 each: salad, fries, milk, juice, jumbo cookie, frozen yogurt. Sentence: 7. Jennifer is saving money to buy a new bike. She doubled the money in her bank account, and then she took out $50. She has $300 left in her account. a) Write an equation to find the amount b) Solve the equation. in her account at the beginning. m 300 m money in Jennifer s account Jennifer had in her bank account. 8. A classroom s length is 3 m less than 2 times its width. The classroom has a length of 9 m. a) Write an equation to find the width of the classroom. Equation: w w l 9 w width b) Solve the equation to find the width of the classroom. Sentence: 546 MHR Chapter 10: Solving Linear Equations
Imagine falling 10 m in 1 s! As an object falls, it picks up more and more speed. A falling object increases its speed by about 10 m for every second it falls. a) Find the speed at which an object hits the ground for different starting speeds and different amounts of time that the object falls. Complete the table. Starting Speed (m/s) Amount of Time the Object Falls (s) Speed at Which It Hits the Ground Starting Speed + (10 Time Falling) (m/s) 0 5 0 + (10 5) 50 5 4 5 + (10 4) 10 1 + (10 ) 0 12 + (10 ) 15 3 b) A stone comes loose from the side of a canyon. It falls with a starting speed of 5 m/s. It hits the water at a speed of 45 m/s. Find the amount of time the stone falls before it hits the water. Speed at which it hits the water starting speed + (10 time falling) + 10t Sentence: 10.2 Math Link MHR 547
10.3 Warm Up 1. Use the opposite operation to solve. a) p 5 20 b) + 9 14 p 5 + 5 20 + p c) 3 15 d) 7 5 d 3 15 3. Draw a model for each equation. Do not solve. a) 7 5 b) 5g + 3 1 4. Multiply or divide. a) 7(2) b) 5( 4) c) 16 2 d) 15 5 5. Subtract. a) 9 ( 3) + Add the opposite. b) 13 6 ( 13) (+6) Rewrite. + Add the opposite. 548 MHR Chapter 10: Solving Linear Equations
10.3 Modelling and Solving Two-Step Equations: a + b c Working Eample 1: Model Equations The elevation of Qamani tuaq, Nunavut, is 1 m less than 1 the elevation of Prince Rupert, B.C. 2 If the elevation of Qamani tuaq is 18 m, what is the elevation of Prince Rupert? a) Write an equation to find the elevation of Prince Rupert. Solution Let p represent the elevation of Prince Rupert. 1 2 the elevation of Prince Rupert 1 2 p The elevation of Qamani tuaq is 1 m less than 1 2 1 2 p 1 The elevation of Qamani tuaq is 18 m, so 1 2 p 1. Nunavut Prince Rupert elevation Qamani tuaq elevation 18 m British Columbia Elevation is the height above sea level. b) Draw a diagram to solve the equation. Check your answer. Solution 1 2 p 1 18 a number that is not connected to a variable The constants in this equation are and 18. p 2 To isolate the variable, add grey square to both sides. positive p 2 The 1 2 circle is equal to 1 2 p grey squares. p 2 1 p 19 2 Multiply both sides by. p 38 The elevation of Prince Rupert is m. 10.3 Modelling and Solving Two-Step Equations: a + b c MHR 549
Solve by modelling each equation. a) 5 7 4 + is a zero pair and represents 1 + ( 1) 0 To isolate the variable, add grey tiles to both sides. 4 1 4 of the circle is equal to white squares. You need 4 equal parts to fill the circle. 1 4 1 4 b) k + 1 4 c) 3 + 5 3 2 550 MHR Chapter 10: Solving Linear Equations
Working Eample 2: Apply the Reverse Order of Operations Kristian Huselius played for the Calgary Flames during the 2006 2007 NHL season. He had 41 more than 1 the number of shots on goal as Jarome Iginla. 2 Huselius had 173 shots on goal. How many shots on goal did Iginla have? a) Write an equation to find the number of shots on goal Jarome Iginla had. Solution Let j represent the number of shots on goal Jarome Iginla had. Huselius had 41 more than 1 2 the number of shots on goal j +. 2 Since Huselius had 173 shots, 2 j + 41. b) Solve the equation to find Jarome Iginla s number of shots on goal. Solution j + 41 173 2 j + 41 173 Subtract from both sides. 2 2 j 2 j 2 2 Multiply both sides by. j Jarome Iginla had shots on goal during the 2006 2007 season. j + 41 2 173 2 + 41 + 41 10.3 Modelling and Solving Two-Step Equations: a + b c MHR 551
Solve by applying the reverse order of operations. a) 6 4 12 6 + 4 + 12 12 12 120 The numerical coefficient of is. 1 120 1 b) 4 3 + 7 k Subtract 3 from both sides. Multiply both sides by 7. 552 MHR Chapter 10: Solving Linear Equations
1. Describe how to isolate the variable when solving 5 n 12 6. 2. Manjit thinks that the first step in solving the equation the equation by 4. He writes: 4 + 7 9 is to multiply both sides of ( 4)+79 ( 4) 4 Is he correct? Circle YES or NO. Give 1 reason for your answer. 3. Write the equation for each diagram and name the constants. A constant is a number that is not connected to a variable. Diagram Equation Constants a) 3 b) -b 2 c) z 5 d) d 7 10.3 Modelling and Solving Two-Step Equations: a + b c MHR 553
g 4. Draw a model for 5 3. Then, solve and check your answer. 5 Model: Solution: g g 5 3 2 5. Solve each equation using the reverse order of operations. Check your answers. a) 2 + 3 m 18 2 + 3 m 18 b) c 8 8 12 m 3 m 3 m 554 MHR Chapter 10: Solving Linear Equations
6. People 18 years old or younger need a certain number of hours of sleep each day. The equation s 12 4 a tells you how many hours of sleep they need. s amount of sleep needed, in hours a age of the person, in years a) If Brian needs 10 h of sleep, how old is he? s 12 4 a b) Natasha is 13 years old. She gets 8 h of sleep a night. Is this enough sleep? s 12 4 a 7. The cost of a concert ticket for a student is $2 less than 1 2 of the cost for an adult. a) Write an epression for the cost of a concert ticket for a student. a the cost for an adult Show $2 less than 1 2 of Cost of student ticket a the adult cost. b) If the cost of a student concert ticket is $5, how much does the adult ticket cost? Equation: Sentence: 10.3 Modelling and Solving Two-Step Equations: a + b c MHR 555
The atmosphere is the air that surrounds Earth. It is made of many different layers. The troposphere is the lowest layer of the atmosphere, where humans live and where weather occurs. Look at the graph. mesosphere ozone layer tropopause stratosphere troposphere 14 000 m a) What pattern do you see in the graph? t 20 10 0 10 20 30 40 50 60 Earth Air Temperature in the Troposphere 2000 4000 6000 8000 10 000 h h b) The equation that models air temperature change in the troposphere is t 15, 154 where t is the temperature, in C, and h is the height, in metres. At what height in the troposphere is the temperature 0 C? h t 15 154 h 0 15 154 h 0 15 Subtract 15 from both sides. 154 h 154 h Multiply both sides by 154. 154 h 1h Divide both sides by 1. h Sentence: 556 MHR Chapter 10: Solving Linear Equations
10.4 Warm Up 1. Fill in the blanks. a) The opposite of adding 4 is b) The opposite of dividing by 6 is c) The opposite of multiplying by 4 is d) The opposite of subtracting 7 is 2. Solve each equation. a) b) 3. Solve. a) 3(4 2) 3( ) b) 2(3 4) 2( ) 4. Solve. a) ( 5) ( 4) b) 4 ( 6) ( 5) + ( ) c) ( 12) + ( 3) d) 7 + ( 8) e) ( 6) ( 3) f) ( 20) 4 g) 7 ( 3) h) ( 8) ( 2) 10.4 Warm Up MHR 557
Name: 10.4 Modelling and Solving Two-Step Equations: a( + b) c Working Eample 1: Model With Algebra Tiles A flower garden is in the shape of a rectangle. The length of the garden is 2 m longer than the length of the shed. The width of the garden is 4 m. The area of the garden is 20 m2. What is the length of the shed? s 2m 4m A 20 m2 a) Write an equation. A 20 m2 Solution 4 s+2 Let s represent the length of the shed. The length of the garden is 2 m longer than the length of the shed s + Area 20 m2, so the equation is 20 4 ( ) or 4(s + 2) 20. Area of a rectangle length width b) Solve the equation. Solution Model the equation 4(s + 2) 20 using algebra tiles. There are 4 groups of s + 2. That means there are 4 black tiles and 8 positive 1-tiles on the left side of the equation. To isolate the variable, add 1-tiles to each side of the equation. white s s s s s s s s equals zero. Remove all the zeros. s Now, there are 4 black tiles on the left and 12 positive 1-tiles on the right. s s Each black tile equals 3 positive 1-tiles, so s The length of the shed is 558 m. MHR Chapter 10: Solving Linear Equations. s
Solve by modelling the equation. a) 2(g + 4) 8 Draw 2 groups of g + 4 on the left side and 8 white 1-tiles on the right. Add 8 white 1-tiles to both sides of the diagram. Remove the zero pairs. Draw the result. Group the tiles into 2 equal groups by circling the equal groups above. Each black tile equals negative 1-tiles. g b) 3(r 2) 3 Draw 3 groups of r 2 on the left side and 3 positive 1-tiles on the right. Add 6 positive 1-tiles to both sides of the diagram. Isolate the variable then draw the result. Group the tiles into 3 equal groups by circling the equal groups above. Each variable tile equals positive 1-tiles. r 10.4 Modelling and Solving Two-Step Equations: a( + b) c MHR 559
Working Eample 2: Solve Equations Kia is making a square quilt with a 4-cm border around it. She wants the quilt to have a perimeter of 600 cm. Find the lengths of the sides of Kia s quilt before she adds the border. s + 8 s 4 cm a) Write an equation. 4 cm Solution Let s represent the length of the side before the border is added. A 4-cm border is added to each end of the side length: s + There are 4 sides to the quilt: perimeter 4(s + 8) P of a square 4s The perimeter is 600 cm, so the equation is 4(s + 8). b) Solve the equation to find the side length of the quilt. Solution distributive property a(b + c) a b + a c when you multiply each term inside the brackets by the term outside the brackets Method 1: Divide First eample: 2( + 3) (2 )+ (2 3) 2 + 6 Method 2: Use the Distributive Property 4(s + 8) 600 4( s + 8) 600 4 4(s + 8) 600 4s + 32 600 4s + 32 600 s + 8 150 s + 8 150 s The side length of the quilt before the border is added is cm. 4 s 4 4 s The quilt dimensions before adding the border are 142 cm 142 cm. 560 MHR Chapter 10: Solving Linear Equations
c) Solve 4( 5) 24. Method 1: Divide First 4( 5) 24 4( 5) 24 4 Divide by 4. + - + - + + - + - - + - 5 ( ) 5 + 5 ( ) + 5 Add to both sides. Method 2: Use the Distributive Property First 4( 5) 24 Multiply and 5 by 4. ( 4) + ( 4)( 5) 24 4 + 24 4 +20 20 24 20 Subtract from both sides. 4 4 4 4 Divide both sides by 4. 4 ( 5) 4 ( 1 5) 4 ( 6) 24 24 10.4 Modelling and Solving Two-Step Equations: a( + b) c MHR 561
a) Solve 2( 3) 12 by dividing first. 2( 3) 12 Divide both sides by 2. ( 3) ( ) 3 + ( ) + Add 3 to both sides. b) Solve 20 5(3 + p) using the distributive property. 20 5(3 + p) Multiply (3 + p) by 5. 20 (5 ) + ( 5 ) Multiply. 20 + 20 ( ) 15 + Subtract 15 from both sides. 20 + ( ) 5p Add the opposite. 5p Divide both sides by 5. 5p p c) Solve 5( 3) 15. 562 MHR Chapter 10: Solving Linear Equations
1. Julia and Chris are solving the equation 6( + 2) 18. a) Julia: Is she correct? Circle YES or NO. Give 1 reason for your answer. First, I subtract 2 from both sides. Then, I divide both sides by 6. b) Chris: Is he correct? Circle YES or NO. Give 1 reason for your answer. I start by dividing 6( + 2) by 6. Then, I subtract 2 from both sides. 2. Model each equation with algebra tiles. a) 3(t 2) 12 b) 6( j 1) 6 c) 2(3 + p) 8 d) 14 7(n 2) 10.4 Modelling and Solving Two-Step Equations: a( + b) c MHR 563
3. Solve the equation modelled by each diagram. Check your answers. a) b) 4. Solve each equation by dividing first. Check your answers. a) 6(r + 6) 18 b) 4(m 3) 12 ( r + ) 6 6 18 r + 6 r + 6 3 r m 564 MHR Chapter 10: Solving Linear Equations
5. Solve each equation using the distributive property. a) 21 3(k + 3) 21 3(k + 3) Multiply k + 3 by 3. 21 + 21 + Subtract 9 from both sides. k Divide both sides by 3. k b) 40 4(n 3) 40 4(n 3) Multiply n and 3 by 4. 40 ( ) + 12 40 ( ) + 12 Subtract 12 from both sides. ( ) 28 4n Divide both sides by 4. n c) 8( 3) 32 d) 3(1 + g) 27 10.4 Modelling and Solving Two-Step Equations: a( + b) c MHR 565
6. An old fence around Gisel s tree is shaped like an equilateral triangle. Gisel wants to build a new fence. She wants to make each side 7 cm longer. She wants the perimeter to be 183 cm. a) Write an equation for this problem. f length of fence before adding 7 cm the sides are the same length The length, f, with 7 cm added Since all 3 sides are equal, the equation is 3( f + 7) perimeter b) Solve the equation to find the length of each side of the old fence. The old fence measures along each of its sides. 7. The formula E 125(t 122) shows the amount of energy a hiker needs each day on a hike. E is the amount of food energy, in kilojoules (kj), and t is the outside temperature, in degrees Celsius. a) If the outside temperature is 20 C, how much food energy will the hiker need each day? Does 20 replace E or t? Sentence: b) If a hiker uses 16 000 kj of food energy, what is the outside temperature? Does 16 000 replace E or t? Sentence: 566 MHR Chapter 10: Solving Linear Equations
In some jobs, people work the night shift. In other jobs, people work alone or with dangerous materials. Sometimes you get paid more money when you have to work under certain conditions. This increase in your pay is called a premium. a) A worker makes r dollars per hour. His boss asks him to work the night shift and offers him a premium of $2 more per hour. What epression represents his new pay? b) If he works 6 h, the epression 6(r + 2) represents his pay. What is the epression if he works for 8 h? c) The worker makes $184 for working an 8-h night shift. Write an equation for this problem. d) How much does he make per hour when he works the night shift? First solve for r. Then add $2. The worker makes per hour on the night shift. e) Research 1 job that pays an hourly or monthly wage plus a premium. Briefly describe the job and eplain why a premium is paid for the job. 10.4 Math Link MHR 567
10 Chapter Review Key Words For #1 to #7, choose the word from the list that goes in each blank. variables distributive property equations constants numerical coefficients opposite operations linear equations 1. Letters that represent unknown numbers are called. 2. are made up of 2 epressions that are equal to each other. 3. Multiplication and division are of each other. 4. Numbers that are attached to a variable by multiplication are called 5. 5(b + 3) 5 b + 5 3 is an eample of how you use the 6. In the equation 2 7 5, both 7 and 5 are. 7. Equations that, when graphed, result in points that lie along a straight line are called... 10.1 Modelling and Solving One-Step Equations: a b, a b, pages 528 536 8. Solve by inspection. a) 6r 18 6 18 r b) 5 p 3 5 3 p 568 MHR Chapter 10: Solving Linear Equations
9. Solve the equation modelled by each diagram. Check your answers. a) b) n 1 1 4 10. Solve each equation using the opposite operation. a) 5t 15 5t t 15 b) a 2 7 a 2 7 a 10.2 Modelling and Solving Two-Step Equations: a + b c, pages 538 547 11. Write and solve the equation modelled by the diagram. Check your answer. c c c Chapter Review MHR 569
12. Solve each equation using opposite operations. a) 3t + 8 20 b) 5j 2 12 3t + 8 20 3t 3t t 13. Zoë has a collection of CDs and DVDs. The number of CDs is 3 fewer than 4 times the number of DVDs. Zoë has 25 CDs. a) Write an equation for this situation. Let d represent the DVDs. number of CDs 3 fewer than 4 times the DVDs: b) Solve the equation. Zoë has DVDs. 10.3 Modelling and Solving Two-Step Equations: + b c, pages 549 556 a 14. Solve the equation modelled by the diagram. 1 1 1 1 1 j 4 Diagram after isolating the variable: Equation: j 570 MHR Chapter 10: Solving Linear Equations
15. Solve. Check your answers. a) d 13 8 3 b) 17 4 + 2 d 13 + 8 + 3 d 3 d 16. An airplane ticket is on sale for $350. The sale price is one third of the regular price, plus $100 in taes. a) Write an equation to represent this situation. Let r the regular price. Sale price 1 of the regular price + $100 3 b) What is the regular price of the airplane ticket? Solve the equation. Sentence: Chapter Review MHR 571
10.4 Modelling and Solving Two-Step Equations: a( + b) c, pages 558 567 17. Solve the equation modelled by each diagram. Check your answers. a) r r r b) w w 18. Solve. Check your answers. a) 6(q 13) 24 b) 2(g + 4) 14 6( q 13) 24 q 572 MHR Chapter 10: Solving Linear Equations
19. Solve using the distributive property. a) 5(w 4) 10 b) 4(m 3) 12 5(w 4) 10 + 10 + 10 10 + ( ) w 20. Each side of a square is decreased by 3 cm. The perimeter of the new square is 48 cm. What is the length of each side of the original square? a) Write an equation. The length of the each side of the original square is s. length of the each side after decreasing it by 3 cm perimeter of the new square Since all 4 sides are equal, the equation is 4( ). b) Solve the equation to find the length of each side of the square. Sentence: Chapter Review MHR 573
10 Practice Test For #1 to #4, circle the best answer. 1. What is the solution to 12? 3 A 36 B 4 C 4 D 36 2. What is the solution to 5n + 6 4? A n 10 B n 5 C n 2 D n 2 3. Which of these equations has the solution p 6? A C p p 4 2 B + 4 2 3 3 p 3 4 2 D p 3 + 4 2 4. Wanda solved the equation 4( 3) 2 like this: 4( 3) 2 Step 1 4 12 8 Step 2 4 20 Step 3 5 In which step did Wanda make her first mistake? A Step 1 B Step 2 C Step 3 D No mistake was made. Complete the statements in #5 and #6. 5. The opposite operation of division is. 6. The solution to 4(y + 5) 24 is y. 574 MHR Chapter 10: Solving Linear Equations
Short Answer 7. Dillon used algebra tiles to model a problem. a) What equation is modelled? b) Using the algebra tiles, what is the first step that Dillon should take to solve the equation? 8. a) Draw algebra tiles to model the equation 3 + 5 7. b) Draw the tiles you need to make zero pairs. Eplain why you need those tiles. c) To solve for, divide both sides of the equation by the numerical coefficient. Eplain why you would divide by this number. d) Solve the equation. Practice Test MHR 575
9. Solve each equation. Check your answers. a) 4 48 b) t 5 8 c) 2k 6 12 d) 12 4( 2) 576 MHR Chapter 10: Solving Linear Equations
10. Beth would like to put a 2-m-wide grass border around a square garden. The perimeter of the outside of the border is 44 m. 2 m a) Write an equation for this situation. Let s be the length of the side of the garden. Perimeter of large square 4 length 4 (s + ) s + s b) Find the length of each side of the garden. Solve your equation. 11. The formula for the perimeter of a rectangle is P 2(l + w). P is the perimeter, l is the length, and w is the width. Find the length of the rectangle if P 14 cm and w 3 cm. w 3 cm Formula P 2(l + w) Equation 2(l + ) Solve Use the distributive property or divide first. l Sentence: Practice Test MHR 577
How are linear equations used in everyday life? For each linear equation in the table: Identify what each variable, numerical coefficient, and constant represents. Solve each equation. Show your steps on a separate sheet of paper. Name a job or a real-life situation where each equation could be used. Equation Type Equation Identify the Variables (V), Coefficients (CO), and Constants (C) Solve the Equation Job or Real-Life Situation Eample: a b 15t 45 t : V 15: CO 45: C 15t 45 15 3 45 t 3 Finding the time it took to finish a race. a b 3d 27 a b p 8 5 a + b c 2c + 7 19 a + b c v 3 11 7 a( + b) c 6( w + 5) 36 578 MHR Chapter 10: Solving Linear Equations
Use the clues to identify key words from Chapter 10. Then, write them in the crossword puzzle. Across 6. In 5d + 4 39, 5 is the. 7. When solving an equation such as 6a 4 26, use the to isolate the variable. 8. In the equation 7 m 6, m is the. Down 1. In the equation 5w + 1 t, 1 is a. 2. 5(s + 2) 5s + 10 uses the property. 3. Two mathematical epressions joined by an equal sign is an. 4. When graphed, a equation results in points along a straight line. 5. When solving 6a 72, you need to the variable. 1 2 3 5 4 6 7 8 Key Word Builder MHR 579
Math Games Rascally Riddles What do mathematicians use to shampoo carpets? To answer the riddle, solve each of the equations. Replace each bo with a letter. The letter you put in the bo will be the variable that has the same solution as the number above the bo. The first one has been done for you. 4 2 4 4 6 4 3 c c 3 2 6 7 3 1 2 1 3 2c 8 2c 8 2 2 c 4 5o 1 9 e r 2 3 4 + 3 2 4i 3 1 3(s 5) 6 l 3 + 4 2 4t + 7 5 2(n + 1) 0 u 7 1 580 MHR Chapter 10: Solving Linear Equations
Challenge in Real Life Earth s Core Earth is made up of layers. The deeper layers are hotter because the temperature inside Earth increases the deeper you go. The table shows how the temperature increases with depth. Layer Depth (km) Temperature ( C) Crust 90 870 Mantle 2921 3700 Outer core 5180 4300 Inner core 6401 7200 1. On a separate sheet of grid paper, graph the data from the table. grid paper coloured pencil Crust Mantle Outer Core 3700 C 870 C 2. Use the formula T d + 870 to find the approimate temperature at any depth where d is the depth, in km, and T is the temperature in ºC. a) Calculate the approimate temperature if the depth is 3400 km. T d + 870 T + 870 b) Check your solution by graphing it on your graph from #1. Does this point lie in a line with the other points? Circle YES or NO. Inner Core 7200 C Centre of Earth 4300 C T The temperature would be at a depth of 3400 km. c) If the temperature is approimately 9000 ºC, how deep are you beneath Earth s surface? T d + 870 d + 870 d) Using a different coloured pencil, check your solution by graphing it on your graph from #1. Does this point lie in a line with the other points? Circle YES or NO. The depth is if the temperature is approimately 9000 ºC. Challenge in Real Life MHR 581
Answers Get Ready, pages 524 525 1. a) y 12 b) y 0 2. a) n + 7 12 b) 3 11 c) 4 28 d) 9 9 3. a) subtract b) subtract c) add d) subtract 4. a) j 4 b) t 8 Math Link Jim s code is 71, 17, 5, 62, 26, 17, 56. 10.1 Warm Up, page 527 1. a) 3 + 6 9 b) 8 5 c) 5 d) 3 6 2. a) + b) + 3. a) 30 b) 24 c) 28 d) 4 e) 5 f) 3 4. a) 3 b) 2 c) 11 d) 5 e) 6 f) 5 g) 40 h) 18 10.1 Modelling and Solving One-Step Equations: a b, a b, pages 528 536 Working Eample 1: Show You Know a) 2 b) y 10 Working Eample 2: Show You Know a) b 9 b) f 2 Working Eample 3: Show You Know d 15 Communicate the Ideas 1. a) 5 15 b) 2. Answers may vary. Eample: Raj multiplied both sides by 9, but he should have used +9. Practise 3. a) 3t 6 b) 4. a) j 6 b) n 5 w 4 c) 2 m 3 5. a) s 3 b) j 12 c) t 24 d) 90 Apply 6. a) 26 km b) 39 km c) 130 km d) 24 L 7. a) 4 b) 4t 72 c) 18 cm Math Link d) Answers may vary. Eample: 2 d) 2n 10 Height of Drop, Height of First Bounce, Trial h b k b h 20 3.0 0.15 1 18 2.7 0.15 2 16 2.4 0.15 3 10 1.5 0.15 4 5 0.8 0.15 e) Answers may vary. Eample: All the numbers in the last column are the same. f) b 0.15h 582 MHR Chapter 10: Solving Linear Equations 10.2 Warm Up, page 537 1. a) 3 4 b) 3 2 4 2. a) 11 b) 9 3. a) 3 b) 6 c) 16 d) 21 4. a) 40 b) 6 5. a) +3 b) +7 c) 13 d) +9 e) 4 f) +4 g) 12 h) 10 10.2 Modelling and Solving Two-Step Equations: a + b c, pages 538 547 Working Eample 1: Show You Know a) n 2 b) p 2 Working Eample 2: Show You Know a) 5 + b) r 5 Working Eample 3: Show You Know a) 4 b) g 8 Communicate the Ideas 1. a) b) 5 grey; You add 5 to both sides to get the -tiles alone. c) 3; You divide by 3 to get 1 tile. Practise 2. a) 3 b) t 7 3. a) 1 b) h 2 4. r r r Second Operation to Equation First Operation to Solve Solve 4r 2 14 Add 2 to each side. Divide each side by 4. 22 10 + 2n Add 10 to each side. Divide each side by 2. 53 9k 1 Add 1 to each side. Divide each side by 9. 3 3 9 Subtract 3 from each side. Divide each side by 3. 5. a) r 2 b) m 4 Apply 6. 3 toppings 7. a) 2m 50 300 b) $175 8. a) 2w 3 9 b) 6 m r Math Link a) b) 4 s Speed at Which It Hits the Ground(m/s) 50 45 20 120 45
10.3 Warm Up, page 548 1. a) p 25 b) 5 c) 5 d) d 35 3. a) b) g g g g g Practise 2. a) t b) t t j j j j 4. a) 14 b) 20 c) 8 d) 3 5. a) 2 b) 19 10.3 Modelling and Solving Two-Step Equations: a + b c, pages 549 556 Working Eample 1: Show You Know a) 8 b) 15 c) k 16 Working Eample 2: Show You Know a) 120 b) k 49 Communicate the Ideas 1. Add 12 to both sides. Multiply both sides by 5. 2. NO. The first step is to subtract 7 from both sides. Practise 3. a) 3 2 5; 2, 5 b) 3 2 b 6; 6, 3 4. c) 5 z + 7 4; 7, 4 d) 3 7 d 8; 8, 3 g 2 g 16 1 1 1 1 1 5. a) m 48 b) c 32 Apply 6. a) 8 years old b) No, she needs 8.75 hours of sleep a night. 7. a) c 1 a 2 b) $14 2 Math Link a) For every 1000-m increase in height, the air temperature drops about 7 C. b) 2310 m 10.4 Warm Up, page 557 1. a) subtracting 4 b) multiplying by 6 c) dividing by 4 d) adding 7 2. a) 2 b) 3 3. a) 6 b) 24 4. a) 1 b) 0 c) 15 d) 1 e) 2 f) 5 g) 21 h) 6 10.4 Modelling and Solving Two-Step Equations: a( + b) c, pages 558 567 Working Eample 1: Show You Know a) g 8 g b) r r 3 Working Eample 2: Show You Know a) 3 b) p 7 c) 0 Communicate the Ideas 1. a) NO. She must divide by 6 first. b) NO. He should divide both sides by 6. c) p d) 3. a) 6 b) 4 4. a) r 9 b) m 6 5. a) k 4 b) n 7 c) 7 d) g 26 Apply 6. a) 3(f + 7) 183 b) 54 cm 7. a) 17 750 kj b) 6 C Math Link a) h(r + 2) b) 8(r + 2) c) 184 8(r + 2) d) $23 e) Answers will vary. Eample: Nurses are paid a premium if they work shifts that are longer than 12 hours. Chapter Review, pages 568 573 1. variables 2. equations 3. opposite operations 4. numerical coefficients 5. distributive property 6. constants 7. linear equations 8. a) r 3 b) p 15 9. a) 3 b) n 8 10. a) t 3 b) a 14 11. 3c + 2 5; c 1 12. a) t 4 b) j 2 13. a) 25 4d 3 b) 7 14. 2 4 j 3; j 4 15. a) d 15 b) 42 16. a) 3 r + 100 350 b) $750 17. a) r 2 b) w 5 18. a) q 17 b) g 3 19. a) w 2 b) m 6 20. a) 48 4( 3) b) 15 cm p Practice Test, pages 574 577 1. D 2. C 3. C 4. A 5. multiplication 6. 1 7. a) 2( 4) 6 or 2 8 6 b) Add 8 positive tiles to both sides or divide both sides by 2 or multiply 2 by and 4. j j n n n n n n n Answers MHR 583
8. a) b) You need zero pairs to isolate the 3 -tiles. c) 3; You divide by 3 to isolate. d) 4 9. a) 12 b) t 40 c) k 9 d) 5 10. a) 44 4(s + 4) b) 7 m 11. 4 cm Wrap It Up!, page 578 Equation Type Equation a b 3d 27 d: V 3: CO 27: C a b p 8 5 p: V 8: CO 5: C a + b c 2c + 7 19 c: V 2: CO 19: C Identify the Variables, Coefficients, and Constants a + b c v 3 11 7 v: V 3: CO 11 and 7: C a( + b) c + + 6(w + 5) 36 w: V 6: CO 5 and 36: C Solve the Equation d 9 p 40 c 6 v 57 w 1 Job or Real- Life Situation Answers will vary. Eample: Finding the force applied to a spring. Finding the temperature of one place compared to somewhere else. Finding the wind speed of one place compared to another. Finding the elevation of one place compared to another. Finding the length of a square when you know the area and how the length compares to the width. Key Word Builder, page 579 Across 6. numerical coefficient 7. opposite operation 8. variable Down 1. constant 2. distributive 3. equation 4. linear 5. isolate Math Games, page 580 e 6, i 1, l 6, n 1, o 2, r 4, s 3, t 3 u 7 Challenge in Real Life, page 581 1. Depth vs. Temperature T 9000 8000 7000 6000 5000 4000 3000 Temperature (ºC) 2000 1000 6000 4000 2000 0 d Depth (km) 2. a) 4270 ºC b) NO c) 8130 m d) YES 584 MHR Chapter 10: Solving Linear Equations