Solving Linear Equations - Mr. Mubashir
CHAPTER10Solving Linear EquationsGET READY524Math Link52610.1 Warm Up52710.1 Modelling and Solving One-StepxEquations: ax b, ba52810.2 Warm Up53710.2 Modelling and Solving Two-StepEquations: ax b c53810.3 Warm Up54810.3 Modelling and Solving Two-StepxEquations: b ca54910.4 Warm Up55710.4 Modelling and Solving Two-StepEquations: a(x b) c558Chapter Review568Practice Test574Wrap It Up!578Key Word Builder579Math Games580Challenge in Real Life581Answers582
Name:Date:Substituting Values into EquationsTo substitute values into an equation:Step 1: Replace the variable.A letter that representsan unknown numberFind the value of y when x 5.y 2(x – 4) 3 Substitute.y 2(5 – 4) 3 Brackets first.y 2(1) 3Multiply.y 2 3Add.y 5Step 2: Follow the order of operations: brackets first multiply and/or divide in order from leftto right add and/or subtract in order from left toright1. Find the value of y when x 4.a) y 2(x – 1) 6y 2(4 – 1) 6y 2(b) y (3 x – 7) 4y (3 ) 6y y ( 6– 7) 4y y – 7) 4 4y Modelling and Solving One-Step EquationsTo solve a word problem, change the words into symbols, letters, and numbers to makean equation.Rewrite asA number increased by 5 is 8.4 times a number is 24.4x 24n 5 84 x 24To solve an equation, use the opposite operation to get x alone on 1 side of the equation.opposite operation an operation that undoes another operation – and are opposite operations, and are opposite operationsExamples:n 5 8n 5–5 8–5n 5 – 5 8–5n 3524What you do to 1 side,do to the other.MHR Chapter 10: Solving Linear Equations4x 244 x 24 44x 6
Name:Date:2. Write each sentence as an equation.a) Seven more than a number is twelve.n b) Three less than a number is eleven.x c) Four times a number is twenty-eight. d) When a number is divided by nine, theresult is nine.Solving Two-Step EquationsTo solve a 2-step equation, find the value of x.2x 3 72x 3 – 3 7 – 32x 3 – 3 7 – 32x 42x 4 22x 2Subtract to undo addition.Get x alone on1 side of theequation.Divide to undo multiplication.3. For each equation, circle the first operation you should do to both sides.a) 2n 4 18b) 3x 5 17Add 4 or subtract 4.Add 5 or subtract 5.c) 8y – 70 94d) 27 7q 6Add 70 or subtract 70.Add 6 or subtract 6.4. Solve the equations.a) 5j 9 295j 9 –b) 2t – 2 14 29 –5j 5j j Get Ready MHR 525
Name:Date:change a wordinto a codeModelling EquationsYou can use equations to encrypt a password.equation 2 expressions joined with an equal sign examples: x – 5 8 or 2y 3 15Jim’s password is weather. He gives each letter in the alphabet a number value.1 a10 j19 s2 b11 k20 t3 c12 l21 u4 d13 m22 v5 e14 n23 w6 f15 o24 xThe number sequence for the password weather is:w237 g16 p25 ye5a18 h17 q26 zt209 i18 rh8e5Jim uses the equation y 3x 2 to encrypt the password.This gives each letter in the password a different number value.Find the new values of the letters.w 23e 5y 3(23) 2y 3(y 69 2y y y t h Jim’s code for weather is 71526MHR Chapter 10: Solving Linear Equationsa 1) 2 2r .r18
Name:10.1 Warm Up1. Write an equation for each diagram.a)xxx xrepresents xrepresents 1represents –1b)c)x Date: xd)xxx 2. Show each equation using algebra tiles.a) 3x 5 –1b) 2x 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 boxes.a) 4 7b) 8 10c)–5 6d)–1 4e) 2 12f) 3 15g) 4 10h) 6 310.1 Warm Up MHR 527
Name:Date:10.1 Modelling and Solving One-Step Equations: ax b,x balinear equation an equation whose points on a graph lie along a straight linec examples: y 4x, d , 5w 1 t2numerical coefficient a number that multiplies the variable example: 4k 3numerical coefficientvariable a letter used to represent an unknown number example: 2y 3variableconstant a number that is not connected to a variable example: 2x 5 –3constantWorking Example 1: Solve an Equationrepresents a positive variable.Solve each equation.represents a negative variable.represents 1.represents –1.a) 3x –12SolutionMethod 1: Solve by InspectionWhat numbertimes 3 equals–12?3x –123 –12Method 2: Solve Using Models andDiagramsUse algebra tiles.xx3 4 –12So, x x3 black tiles 3x12 white tiles –123 black tiles 12 white tilesHow many white tiles 1 black tile?So, x 528MHR Chapter 10: Solving Linear Equations.
Name: Date:r 7 2b)SolutionMethod 1: Solve by Inspectionr 7 2 214 2What number dividedby –2 equals –7? 7 7Addwhite tiles to show –7.-r2.Check:Left Sider 2 1 whole circle –rDivide the circle into 2 parts.Shade half the circle.r –2-r2So, x Method 2: Solve Using Models andDiagramsRight Side–7 9 -1 -1 -1 -1 -1 -1 -1 r, so multiply the2amount of shaded circle by 2 to represent –r.Half the circle showsThen, multiply the number of white squaresby 2 so that the equation is balanced. 29 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1-rwhite squares represent –14.So, –r .To make r positive, multiply both sides ofthe equation by –1.–1(–r) –1(–14)So, r .10.1 Modelling and Solving One-Step Equations: ax b,x b MHR 529a
Name: Date:Solve each equation.Use inspection, models, or diagrams.Check your work.Remember, - - - - -b)a) 4x –8Check:Left Side4xRight Side–8 4 y 52Check:Left Sidey2 530MHR Chapter 10: Solving Linear Equations2Right Side5
Name: Date:Working Example 2: Divide to Apply the Opposite OperationSimone uses a spring to do an experiment.The equation F 12d shows how much force the spring has.F force in newtons, Nd distance the spring stretches, in cmSimone pushes the spring together with a force of 84 N.What distance is the spring compressed?pressed togetherSolutionF 12d–84 12d 84 12dSubstitute –84 for F.The opposite of 12 is12.Check:Left SideC 84 12 ?–84 9Right Side12d 12 (–7) dThe spring is compressed 9cm.The spring was pushed in,so d is negative.Solve by applying the opposite operation. Check your answer.–5b –45a) 5b b) 6f –12 45b Check:Left Side–5bRight SideCheck:Left SideRight Side–45 –5 10.1 Modelling and Solving One-Step Equations: ax b,x b MHR 531a
Name:Date:Working Example 3: Multiply to Apply the Opposite OperationIn January, the average afternoon temperature in Edmonton isafternoon temperature in Yellowknife.The average afternoon temperature in Edmonton is –8 C.What is the average afternoon temperature in Yellowknife?1the average3SolutionLet t represent the average afternoon temperature in Yellowknife.11tthe average afternoon temperature in Yellowknife t or333tThe equation to model the problem is –8.3t –83t 3 8 33The opposite of 3 is3.t The average afternoon temperature inYellowknife is C.Check:Left Sidet3 24 3 Right Side–899Solve by applying the opposite operation. Check your answer.d 3 5d 3 5d Check:Left Sided 5 532MHR Chapter 10: Solving Linear Equations 5Right Side3
Name:Date: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 equationn –4n –4.99n X (–9) -4 X (–9)9n 36Raj’s solution is wrong. Explain where he made his mistake.3. Write the equation modelled by each diagram.a)tt trepresents 1represents –1b) -1 -1 -1 -1-w2t c)m -1 -13d)-n-n m 310.1 Modelling and Solving One-Step Equations: ax b,x b MHR 533a
Name:Date:4. Solve by inspection.–2j 12a)–2 5n –25b)5 12 –25j n 5. Solve each equation using the opposite operation. Check your answers.4s –12a)4sb) –36 –3j 12 s Check:Left SideRight Sidet –83c) Check:Left Sidet –8 39 d) 9 t Check:Left Side534Right Sidex 10x 10 xRight SideMHR Chapter 10: Solving Linear EquationsCheck:Left SideRight Side
Name:Date: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 lives312 km away.How many litres of gas will he need?Use the equation 13x 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: ax b,x b MHR 535a
Name:Date: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 experiment 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 next 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.TrialHeight of Drop,hHeight of First Bounce,bk Example20 cm3 cmk bh320 0.151234e) What do you notice about the numbers you recorded in the last column?f) Write an equation that shows the results of your experiment.Substitute your value for k into the equation.b 536hMHR Chapter 10: Solving Linear Equations
Name: Date:10.2 Warm Up1. Write the equation for each diagram.a)xb) xxx 2. Evaluate.a) b 6, when b 5b) s – 3, when s 12b 6 6 3. Find the missing number by inspection.a) 5 15b) 4 24 3 7 2 8c)4. Multiply.d)3 73(2) 3 2 6a) 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 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15a) (–5) ( 8) b) ( 11) (–4) c) (–6) (–7) d) (–3) ( 12) ( 8) – (–6)e) ( 8) ( 6) g) (–9) – ( 3)(–5) – (–9)f) Add the opposite (–5) () h) ( 2) – ( 12)10.2 Warm Up MHR 537
Name: Date:10.2 Modelling and Solving Two-Step Equations: ax b cWorking Example 1: Modelling With a Balance ScaleSolve the equation 23 4k 3.SolutionYou can use a balance scale to model this equation.The numerical coefficient iskkkk, so draw 4 black tiles.the number thatmultiplies the variableIsolate the variable by removinggrey tiles from the right.To keep things balanced, removegrey tiles from the left.kkkkThere arek blocks on the right side of the scale.There aregrey blocks on the left side of the scale.To balance the scale, each k block must have a mass ofSo, k .Check:Left Side23 9grey blocks.Right Side4k 3 4(5) 3 20 3 538MHR Chapter 10: Solving Linear Equations9
Name: Date:Use the balance scale to solve each equation.Show your steps, then check your answer.a) 6n 6 18b) 13 9 2pSo, n .Check:Left Side6n 6 6( ) 6 6Right Side18Check:Left Side13Right Side9 2p 9 2() 9 10.2 Modelling and Solving Two-Step Equations: ax b c MHR 539
Name: Date:Working Example 2: Model With Algebra TilesA 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?unknown value number ofhours an elephant sleepsa) Write an equation for this situation.SolutionLet 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.SolutionThe numerical coefficient iseblack tiles.the number thatmultiplies the variable e, so drawTo isolate the black tiles on 1 side of the equal sign, addboth sides.eThe negative 1-tile and positive1-tile on the left side equal 0. eblack tile is equal toeegrey tiles. e 4So, an elephant sleeps540grey tile toh a day.MHR Chapter 10: Solving Linear Equations
Name: Date:Solve each equation using algebra tiles.a) 2x 4 –6Fill 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 towhite tiles.x b) 3r – 2 13r 10.2 Modelling and Solving Two-Step Equations: ax b c MHR 541
Name: Date:Working Example 3: Apply the Opposite OperationsSolve 4w 3 19 by applying the opposite operation. and – and SolutionTo isolate a variable, follow the reverse order of operations.4w 3 19Subtract 3 from both sides of the equation.4w 3 – 3 19 – 34w Divide both sides by 4.4x 44w Check using algebra tiles:Draw algebra tiles to represent the equation.wwwwIsolate the black tiles.Subtractgrey 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 ofgrey 1-tiles.w wwww wwww Solve by applying the opposite operation.–3x – 7 5a)–3x – 7 5 –3x 3x 4 2g –12b)4– 2g –12 – ()2g –12 ()2g x g 542MHR Chapter 10: Solving Linear Equations
Name:Date:1. a) Draw algebra tiles to model 3x – 5 16.Isolate means to getthe variable alone.b) To isolate x, addtiles 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 x, divide both sides of the equation byGive 1 reason why you need to divide by this number.2. Solve each equation modelled by the algebra tiles.xa)xxb)tt xx t 10.2 Modelling and Solving Two-Step Equations: ax b c MHR 543
Name:Date:3. Solve the equation modelled by each balance scale. Check your solution.a) 3x 8 11b) 13 4h 5Remove 8 grey tiles from each side.xxxx 544MHR Chapter 10: Solving Linear Equationsh
Name:Date:4. Complete the table.EquationFirst Operation to Solve4r – 2 14Add to each side.Second Operation to SolveDivide both sides by .–22 –10 2n53 –9k – 13 – 3x –95. Solve each equation and check your answer.6r 6 18a)6r 6 –b) 4m – 2 14 18 –6r 6r r Check:Left Side6r 6 6( Right Side18Check:Left SideRight Side) 6 6 10.2 Modelling and Solving Two-Step Equations: ax b c MHR 545
Name:6. You buy lunch at Sandwich Express.A sandwich costs 4. Each extra topping costs 2.You have 10. Use the equation 2e 4 10 to findhow many extra toppings you can get if you spendall of your money.Date:MENUYour choice of extras,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 amountin her account at the beginning.b) Solve the equation.m 300m money inJennifer’s accountJennifer hadin 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.l 9a) Write an equation to find the width of the classroom.wEquation:w– w widthb) Solve the equation to find the width of the classroom.Sentence:546MHR Chapter 10: Solving Linear Equations
Name:Date: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 differentamounts of time that the object falls.Complete the table.Speed at Which It Hits the Ground Starting Speed (10 Time Falling)(m/s)StartingSpeed (m/s)Amount of Timethe Object Falls(s)0554101 (10 ) 012 (10 ) 1530 (10 5) 505 (10 4) 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) 10tSentence:10.2 Math Link MHR 547
Name:Date:10.3 Warm Up1. Use the opposite operation to solve.p – 5 20a)b) x 9 14p – 5 5 20 p d) 7 3x 15c)3x d515x 3. Draw a model for each equation. Do not solve.a) x – 7 5b) 5g 3 –14. Multiply or divide.a) 7(2) c)b) 5(–4) 16 2d) 15 55. Subtract.9 – (–3)a) b) –13 – 6 (–13) – ( 6)Rewrite. Add the opposite.Add the opposite. 548MHR Chapter 10: Solving Linear Equations
Name:Date:x b ca10.3 Modelling and Solving Two-Step Equations:Working Example 1: Model Equations1the elevation of Prince Rupert, B.C.2If the elevation of Qamani’tuaq is 18 m, what is the elevation of Prince Rupert?The elevation of Qamani’tuaq, Nunavut, is 1 m less thana) Write an equation to find the elevation of Prince Rupert.NunavutSolutionQamani’tuaqelevation 18 mLet p represent the elevation of Prince Rupert.11“ the elevation of Prince Rupert” p22“The elevation of Qamani’tuaq is 1 m less thanThe elevation of Qamani’tuaq is 18 m, soPrince Rupertelevation 䊏BritishColumbia11” p–1221p–1 2Elevation is the heightabove sea level.b) Draw a diagram to solve the equation. Check your answer.Solutiona number that is notconnected to a variable1p – 1 182 1 1 1 1 1 1The constants in this equation areTo isolate the variable, addgrey square to both sides.Theand 18.positive1circle is equal to2grey squares.1p 2 1p 19 2p 38The elevation of Prince Rupert is-1 1 1 1 1 1 1 1 1 1 1 1 1p2-1p2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1p2 1 1 1 1 1 1 1 1 1 1 1 1Multiply both sides by.m.10.3 Modelling and Solving Two-Step Equations:x b c MHR 549a
Name:Date:Solve by modelling each equation.a)x– 5 –74 is a zero pair andrepresents 1 (–1) 0-1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1To isolate the variable, addgrey tiles to both sides. x41of the circle is equal to4white squares.You need 4 equal partsto fill the circle.1x 4 1x 4 x b)550x 1 –43MHR Chapter 10: Solving Linear Equationsc) 3 k –52
Name:Date:Working Example 2: Apply the Reverse Order of OperationsKristian Huselius played for the Calgary Flames during the 2006–2007 NHL season.1He had 41 more than the number of shots on goal as Jarome Iginla.2Huselius had 173 shots on goal.How many shots on goal did Iginla have?a) Write an equation to find the number of shots on goalJarome Iginla had.SolutionLet j represent the number of shots on goal Jarome Iginla had.1j“Huselius had 41 more than the number of shots on goal” 22jSince Huselius had 173 shots, 41 .2.b) Solve the equation to find Jarome Iginla’s number of shots on goal.Solutionj 41 1732j 41 –2 173 –j 2j2 2 2Subtractfrom both sides.Multiply both sides by.j Jarome Iginla hadCheck:Left Sidej 412 2shots on goal during the 2006–2007 season.Right Side173 9 41 41 910.3 Modelling and Solving Two-Step Equations:x b c MHR 551a
Name:Date:Solve by applying the reverse order of operations. x–6 412a) x–6 12 4 x 12 –x 120 1x120 1 12The numerical coefficient of x isx b) –4 3 k7Subtract 3 from both sides.Multiply both sides by 7.552MHR Chapter 10: Solving Linear Equations.
Name:1. Describe how to isolate the variable when solvingDate:n– 12 6.52. Manjit thinks that the first step in solving the equationthe equation by –4. He writes:x–4x 7 9 is to multiply both sides of 4 ( –4) 7 9 ( –4)Is he correct? Circle YES or NO.Give 1 reason for your answer.A constant is a number that isnot connected to a variable.3. Write the equation for each diagram and name the constants.Diagrama)b)-1x3-1 1 1-1 -1 1 -1 -1-b2c)-1 -1 1 1 1z5Constants 1 1 1 1 1d) Equation 1 1 1 1 1 1 1 1-1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1d710.3 Modelling and Solving Two-Step Equations:x b c MHR 553a
Name:4. Draw a model forDate:g 5 3 . Then, solve and check your answer.5Model:Solution:Check:Left Sideg–52g Right Side35. Solve each equation using the reverse order of operations. Check your answers.m3m 3m3m 32 a)2– 18b)c– 8 –12 8 18 – m Check:Left Side554Right SideMHR Chapter 10: Solving Linear EquationsCheck:Left SideRight Side
Name:Date:6. People 18 years old or younger need a certain number of hours of sleep each day.aThe equation s 12 – tells you how many hours of sleep they need.4s amount of sleep needed, in hoursa age of the person, in yearsa) If Brian needs 10 h of sleep, how old is he?s 12 –b) Natasha is 13 years old. She gets 8 h ofsleep a night. Is this enough sleep?a4s 12 –7. The cost of a concert ticket for a student is 2 less thana41of the cost for an adult.2a) Write an expression for the cost of a concert ticket for a student.a the cost for an adultShow 2 less thanCost of student ticket athe adult cost.1of2b) 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:x b c MHR 555a
Name: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.Date:mesospheretropopauseozone layertropospherestratosphere14 000 mEartha) What pattern do you see in the graph?t Air Temperature in the Troposphere20100 10 20 402000400060008000 10 000h 30 50 60b) The equation that models air temperature change in the troposphere is t 15 –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,154h154h0 15 154t 15 0 15 h154Subtract 15 from both sides. h154 h154Multiply both sides by 154. h 1hDivide both sides by 1. hSentence:556MHR Chapter 10: Solving Linear Equations
Name:Date:10.4 Warm Up1. Fill in the blanks.a) The opposite of adding 4 isb) The opposite of dividing by 6 isc) The opposite of multiplying by –4 isd) The opposite of subtracting 7 is2. Solve each equation.a)b) 3. Solve.3(4 – 2)a) 3(2(3 4)b)) 2( ) 4. Solve.(–5) – (–4)a) (–5) (b) 4 – (–6)) 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:Date:10.4 Modelling and Solving Two-Step Equations: a(x b) cWorking Example 1: Model With Algebra TilesA 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?s2m4mA 20 m2a) Write an equation.A 20 m2Solution4s 2Let 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 widthb) Solve the equation.SolutionModel 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 positive1-tiles on the left side of the equation.To isolate the variable, add1-tiles to each side of the equation.whitess ssssss equals zero.Remove all the zeros.sNow, there are 4 black tiles on the left and12 positive 1-tiles on the right.ssEach black tile equals 3 positive 1-tiles, so s The length of the shed is558m.MHR Chapter 10: Solving Linear Equations.s
Name:Date:Solve by modelling the equation.a) 2(g 4) –8Draw 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 equalsnegative 1-tiles.g b) 3(r – 2) 3Draw 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 equalspositive 1-tiles.r 10.4 Modelling and Solving Two-Step Equations: a(x b) c MHR 559
Name:Date:Working Example 2: Solve EquationsKia 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.4 cms 8sa) Write an equation.4 cmSolutionLet 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 P of a square 4sThere are 4 sides to the quilt: perimeter 4(s 8)The perimeter is 600 cm, so the equation is 4(s 8) .b) Solve the equation to find the side length of the quilt.Solutiondistributive property a(b c) a b a c when you multiply each term inside the brackets by the term outsidethe brackets example: 2(x 3) (2 x ) (2 3) 2x 6Method 2: Use the Distributive PropertyMethod 1: Divide First4(s 8) 6004(s 8) 6004 ( s 8) 44s 32 6006004s 32 – 600 –s 8 150s 8–4s 4 150 –s s The side length of the quilt before theborder is added is5604cm.MHR Chapter 10: Solving Linear EquationsThe quilt dimensions before addingthe border are 142 cm 142 cm.
Name:Date:c) Solve –4(x – 5) 24.Method 1: Divide First–4(x – 5) 24 4 ( x 5)24 4 - - - - -Divide by – 4.x – 5 (–)x – 5 5 (–) 5Addto both sides.x Method 2: Use the Distributive Property First–4(x – 5) 24Multiply x and –5 by –4.(–4x) (–4)( –5) 24–4x 24–4x 20 – 20 24 – 20Subtractfrom both sides.–4x 4 x 44Divide both sides by –4.x Check:Left Side–4 (x – 5) –4 (–1 – 5) –4 (–6) 24 9Right Side24 910.4 Modelling and Solving Two-Step Equations: a(x b) c MHR 561
Name:Date:a) Solve –2(x – 3) 12 by dividing first. 2( x 3) 12Divide both sides by –2.(x – 3) (x–3 ) () Add 3 to both sides.x b) Solve –20 5(3 p) using the distributive property.–20 5(3 p)Multiply (3 p) by 5.–20 (5 –20 ) (5 –20 – () 15 ––20 () 5pSubtract 15 from both sides.Divide both sides by 5.5pp c) Solve 5(x – 3) –15.562 Add the opposite. 5p ) Multiply.MHR Chapter 10: Solving Linear Equations
Name:Date:1. Julia and Chris are solving the equation –6(x 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(x 2) by –6.Then, I subtract 2 from both sides.2. Model each equation with algebra tiles.a) 3(t – 2) 12b) 6( j – 1) –6c) 2(3 p) 8d) 14 7(n – 2)10.4 Modelling and Solving Two-Step Equations: a(x b) c MHR 563
Name:Date:3. Solve the equation modelled by each diagram. Check your answers.a)x xb)x xxCheck:Left SideRight SideCheck:Left SideRight Sidex4. Solve each equation by dividing first. Check your answers.6(r 6) –18a)6 ( r 6) b) 4(m – 3) 12 18r 6 r 6– 3–m r Check:Left Side564Right SideMHR Chapter 10: Solving Linear EquationsCheck:Left SideRight Side
Name:Date:5. Solve each equation using the distributive property.21 3(k 3)a)21 3(k 3)21 –Multiply k 3 by 3.21 –Subtract 9 from both sides. k Divide both sides by 3.k 40 –4(n – 3)b)40 –4(n – 3)40 (40 –Multiply n and –3 by –4.) 12 (–) 12 – (28 Subtract 12 from both sides.) 4nDivide both sides by –4.n c) 8(x – 3) 32d) 3(1 g) 2710.4 Modelling and Solving Two-Step Equations: a(x b) c MHR 565
Name:Date: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.the sides are thesame lengtha) Write an equation for this problem.f length of fence before adding 7 cmThe length, f, with 7 cm added perimeterSince all 3 sides are equal, the equation is 3( f 7) b) Solve the equation to find the length of each side of the old fence.The old fence measuresalong 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 –20replace E or t?Sentence:b) If a hiker uses 16 000 kJ of food energy, what is the outside temperature?Does 16 000replace E or t?Sentence:566MHR Chapter 10: Solving Linear Equations
Name: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 workunder certain conditions.Date:This increase in your payis 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 expression represents his new pay?b) If he works 6 h, the expression 6(r 2) represents his pay.What is the expression 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 makesper hour on the night shift.e) Research 1 job that pays an hourly or monthly wage plus a premium.Briefly describe the job and explain why a premium is paid for the job.10.4 Math Link MHR 567
Name:Date:10 Chapter ReviewKey WordsFor #1 to #7, choose the word from the list that goes in each blank.variablesnu
Solving Linear Equations GET READY 524 Math Link 526 10.1 Warm Up 527 10.1 Modelling and Solving One-Step Equations: ax b, x a b 528 10.2 Warm Up 537 10.2 Modelling and Solving Two-Step Equations: ax b c 538 10.3 Warm Up 548 10.3 Modelling and Solving Two-Step Equations: x a b c 549 10.4 Warm Up 557 10.4 Modelli
9.1 Properties of Radicals 9.2 Solving Quadratic Equations by Graphing 9.3 Solving Quadratic Equations Using Square Roots 9.4 Solving Quadratic Equations by Completing the Square 9.5 Solving Quadratic Equations Using the Quadratic Formula 9.6 Solving Nonlinear Systems of Equations 9 Solving Quadratic Equations
KEY: system of linear equations solution of a system of linear equations solving systems of linear equations by graphing solving systems of linear equations NOT: Example 2 3. 2x 2y 2 7x y 9 a. (1, 9) c. (0, 9) b. (2, 3) d. (1, 2) ANS: D REF: Algebra 1 Sec. 5.1 KEY: system of linear equations solution of a system of .
Section 5.1 Solving Systems of Linear Equations by Graphing 237 Solving Systems of Linear Equations by Graphing The solution of a system of linear equations is the point of intersection of the graphs of the equations. CCore ore CConceptoncept Solving a System of Linear Equations by Graphing Step
1 Solving Linear Equations 1.1 Solving Simple Equations 1.2 Solving Multi-Step Equations 1.3 Solving Equations with Variables on Both Sides 1.4 Rewriting Equations and Formulas Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems arising in ev
EQUATIONS AND INEQUALITIES Golden Rule of Equations: "What you do to one side, you do to the other side too" Linear Equations Quadratic Equations Simultaneous Linear Equations Word Problems Literal Equations Linear Inequalities 1 LINEAR EQUATIONS E.g. Solve the following equations: (a) (b) 2s 3 11 4 2 8 2 11 3 s
Precalculus: Linear Equations Concepts: Solving Linear Equations, Sketching Straight Lines; Slope, Parallel Lines, Perpendicular Lines, Equations for Straight Lines. Solving Linear Equations and Inequalities An equation involves an equal sign and indicates that two expressions have the same v
Lesson 2a. Solving Quadratic Equations by Extracting Square Roots Lesson 2b. Solving Quadratic Equations by Factoring Lesson 2c. Solving Quadratic Equations by Completing the Square Lesson 2d. Solving Quadratic Equations by Using the Quadratic Formula What I Know This part will assess your prior knowledge of solving quadratic equations
Contractors may use a Government employee's Government Purchase Card as long as the Contracting Office authorizes its use and the contractor notifies the Contracting Officer of all purchases made by the contractor during the billing period. Unauthorized Use of the GPC Page 7 of 29 Knowledge Review Computer supplies from a GSA schedule. Food, drinks, clothing, lodging or travel related expenses .
