1 Solving Linear Equations - Big Ideas Learning
11.11.21.31.4Solving Linear EquationsSolving Simple EquationsSolving Multi-Step EquationsSolving Equations with Variables on Both SidesRewriting Equations and FormulasDensity of Pyrite (p. 33)SEE the Big IdeaSki Club (p. 32)Boat (p. 24)Biking (p. 14)Average Speed (p. 6)Mathematicalal Thinking: Mathematically proficient students can apply the mathematicsmathey know to solve problemsarising in everydayay life, society, and the workplace.
Maintaining Mathematical ProficiencyAdding and Subtracting IntegersExample 1(6.3.D)Evaluate 4 ( 12). 12 4 . So, subtract 4 from 12 .4 ( 12) 8Use the sign of 12.Example 2Evaluate 7 ( 16). 7 ( 16) 7 16Add the opposite of 16. 9Add.Add or subtract.1. 5 ( 2)2. 0 ( 13)3. 6 144. 19 ( 13)5. 1 66. 5 ( 7)7. 17 58. 8 ( 3)9. 11 15Multiplying and Dividing IntegersExample 3(6.3.D) Evaluate 3 ( 5).The integers have the same sign. 3 ( 5) 15The product is positive.Example 4Evaluate 15 ( 3).The integers have different signs.15 ( 3) 5The quotient is negative.Multiply or divide. 10. 3 (8)11. 7 ( 9)12. 4 ( 7)13. 24 ( 6)14. 16 215. 12 ( 3)17. 36 618. 3( 4) 16. 6 819. ABSTRACT REASONING Summarize the rules for (a) adding integers, (b) subtracting integers,(c) multiplying integers, and (d) dividing integers. Give an example of each.1
MathematicalThinkingMathematically proficient students display, explain, and justifymathematical ideas and arguments using precise mathematicallanguage in written or oral communication. (A.1.G)Specifying Units of MeasureCore ConceptOperations and Unit AnalysisAddition and SubtractionWhen you add or subtract quantities, they must have the same units of measure.The sum or difference will have the same unit of measure.ExamplePerimeter of rectangle (3 ft) (5 ft) (3 ft) (5 ft)3 ft 16 feet5 ftWhen you add feet,you get feet.Multiplication and DivisionWhen you multiply or divide quantities, the product or quotient will have adifferent unit of measure.ExampleArea of rectangle (3 ft) (5 ft) 15 square feetWhen you multiply feet, you getfeet squared, or square feet.Specifying Units of MeasureYou work 8 hours and earn 72. What is your hourly wage?SOLUTIONdollars per hourdollars per hourHourly wage 72 8 h( per h) 9 per hourThe units on each side of theequation balance. Both arespecified in dollars per hour.Your hourly wage is 9 per hour.Monitoring ProgressSolve the problem and specify the units of measure.1. The population of the United States was about 280 million in 2000 and about310 million in 2010. What was the annual rate of change in population from2000 to 2010?2. You drive 240 miles and use 8 gallons of gasoline. What was your car’s gas mileage(in miles per gallon)?3. A bathtub is in the shape of a rectangular prism. Its dimensions are 5 feet by 3 feet by18 inches. The bathtub is three-fourths full of water and drains at a rate of 1 cubic footper minute. About how long does it take for all the water to drain?2Chapter 1Solving Linear Equations
1.1Solving Simple EquationsEssential QuestionTEXAS ESSENTIALKNOWLEDGE AND SKILLSHow can you use simple equations to solvereal-life problems?A.5.AMeasuring AnglesWork with a partner. Use a protractor to measure the angles of each quadrilateral.Copy and complete the table to organize your results. (The notation m A denotes themeasure of angle A.) How precise are your ALTERMSA conjecture is anunproven statementabout a generalmathematical concept.After the statement isproven, it is called arule or a theorem.QuadrilateralDm A(degrees)m B(degrees)DCm C(degrees)Cm A m B m C m Dm D(degrees)a.b.c.Making a ConjectureWork with a partner. Use the completed table in Exploration 1 to write a conjectureabout the sum of the angle measures of a quadrilateral. Draw three quadrilaterals thatare different from those in Exploration 1 and use them to justify your conjecture.Applying Your ConjectureWork with a partner. Use the conjecture you wrote in Exploration 2 to write anequation for each quadrilateral. Then solve the equation to find the value of x. Usea protractor to check the reasonableness of your answer.a.b.85 c.78 30 100 x 90 80 x 72 x 60 90 Communicate Your Answer4. How can you use simple equations to solve real-life problems?5. Draw your own quadrilateral and cut it out. Tear off the four corners ofthe quadrilateral and rearrange them to affirm the conjecture you wrote inExploration 2. Explain how this affirms the conjecture.Section 1.1Solving Simple Equations3
Lesson1.1What You Will LearnSolve linear equations using addition and subtraction.Solve linear equations using multiplication and division.Core VocabulVocabularylarryUse linear equations to solve real-life problems.conjecture, p. 3rule, p. 3theorem, p. 3equation, p. 4linear equationin one variable, p. 4solution, p. 4inverse operations, p. 4equivalent equations, p. 4Solving Linear Equations by Adding or SubtractingAn equation is a statement that two expressions are equal. A linear equation in onevariable is an equation that can be written in the form ax b 0, where a and b areconstants and a 0. A solution of an equation is a value that makes the equation true.Inverse operations are two operations that undo each other, such as additionand subtraction. When you perform the same inverse operation on each side of anequation, you produce an equivalent equation. Equivalent equations are equationsthat have the same solution(s).PreviousexpressionCore ConceptAddition Property of EqualityWords Adding the same number to each side of an equation producesan equivalent equation.If a b, then a c b c.AlgebraSubtraction Property of EqualityWords Subtracting the same number from each side of an equation producesan equivalent equation.AlgebraIf a b, then a c b c.Solving Equations by Addition or SubtractionSolve each equation. Justify each step. Check your answer.a. x 3 5b. 0.9 y 2.8SOLUTIONa. x 3 5 3Addition Property of EqualityWrite the equation. 3Checkx 3 5? 2 3 5Add 3 to each side.x 2Simplify. 5 5The solution is x 2.b.Subtraction Property of Equality0.9 y 2.8 2.8 2.8 1.9 yWrite the equation.CheckSubtract 2.8 from each side.0.9 y 2.8?0.9 1.9 2.8Simplify.The solution is y 1.9.Monitoring Progress 0.9 0.9 Help in English and Spanish at BigIdeasMath.comSolve the equation. Justify each step. Check your solution.1. n 3 74Chapter 1Solving Linear Equations122. g —3 —33. 6.5 p 3.9
Solving Linear Equations by Multiplying or DividingCore ConceptMultiplication Property of EqualityWords Multiplying each side of an equation by the same nonzero numberREMEMBERproduces an equivalent equation.Multiplication and divisionare inverse operations. If a b, then a c b c, c 0.AlgebraDivision Property of EqualityWords Dividing each side of an equation by the same nonzero numberproduces an equivalent equation.If a b, then a c b c, c 0.AlgebraSolving Equations by Multiplication or DivisionSolve each equation. Justify each step. Check your answer.na. — 35b. π x 2πc. 1.3z 5.2SOLUTIONn — 35a.Multiplication Property of Equality 5 ( )Write the equation. n — 5 ( 3)5Checkn — 3515 ? — 35 3 3Multiply each side by 5.n 15Simplify.The solution is n 15.b. π x 2πDivision Property of EqualityπxπWrite the equation. 2ππ— —Divide each side by π.x 2 Checkπ x 2π?π ( 2) 2πSimplify. 2π 2πThe solution is x 2.c. 1.3z 5.2Division Property of Equality1.3z 5.2— —1.31.3z 4Write the equation.Check1.3z 5.2?1.3(4) 5.2Divide each side by 1.3.Simplify.5.2 5.2The solution is z 4.Monitoring Progress Help in English and Spanish at BigIdeasMath.comSolve the equation. Justify each step. Check your solution.y34. — 65. 9π π xSection 1.16. 0.05w 1.4Solving Simple Equations5
Solving Real-Life ProblemsCore ConceptAPPLYINGMATHEMATICSFour-Step Approach to Problem SolvingMathematically proficientstudents routinely checkthat their solutions makesense in the context of areal-life problem.1.Understand the Problem What is the unknown? What information is beinggiven? What is being asked?2.Make a Plan This plan might involve one or more of the problem-solvingstrategies shown on the next page.3.Solve the Problem Carry out your plan. Check that each step is correct.4.Look Back Examine your solution. Check that your solution makes sense inthe original statement of the problem.Modeling with MathematicsIn the 2012 Olympics, Usain Bolt won the200-meter dash with a time of 19.32 seconds. Writeand solve an equation to find his average speed tothe nearest hundredth of a meter per second.REMEMBERSOLUTIONThe formula that relatesdistance d, rate or speed r,and time t isd rt.1. Understand the Problem You know thewinning time and the distance of the race.You are asked to find the average speed tothe nearest hundredth of a meter per second.2. Make a Plan Use the Distance Formula to writean equation that represents the problem. Thensolve the equation.3. Solve the Problem 200 r 19.32d r tREMEMBERThe symbol means“approximately equal to.”20019.3219.32r19.32Write the Distance Formula.Substitute 200 for d and 19.32 for t.— —Divide each side by 19.32.10.35 rSimplify.Bolt’s average speed was about 10.35 meters per second.4. Look Back Round Bolt’s average speed to 10 meters per second. At this speed,it would take200 m10 m/sec— 20 secondsto run 200 meters. Because 20 is close to 19.32, your solution is reasonable.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com7. Suppose Usain Bolt ran 400 meters at the same average speed that he ran the200 meters. How long would it take him to run 400 meters? Round your answerto the nearest hundredth of a second.6Chapter 1Solving Linear Equations
Core ConceptCommon Problem-Solving StrategiesUse a verbal model.Guess, check, and revise.Draw a diagram.Sketch a graph or number line.Write an equation.Make a table.Look for a pattern.Make a list.Work backward.Break the problem into parts.Modeling with MathematicsOn January 22, 1943, the temperature in Spearfish, South Dakota, fell from 54 Fat 9:00 a.m. to 4 F at 9:27 a.m. How many degrees did the temperature fall?SOLUTION1. Understand the Problem You know the temperature before and after thetemperature fell. You are asked to find how many degrees the temperature fell.2. Make a Plan Use a verbal model to write an equation that represents the problem.Then solve the equation.3. Solve the ProblemWordsNumber of degreesTemperatureTemperature the temperature fellat 9:27 a.m.at 9:00 a.m.VariableLet T be the number of degrees the temperature fell.Equation 4 54 4 54 TTWrite the equation. 4 54 54 54 T 58 TSubtract 54 from each side.Simplify.58 TDivide each side by 1.The temperature fell 58 F.REMEMBERThe distance between twopoints on a number line isalways positive.4. Look Back The temperature fell from 54 degrees above 0 to 4 degrees below 0.You can use a number line to check that your solution is reasonable.58 8 4048 12 16 20 24 28 32 36 40 44 48 52 56 60Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com8. You thought the balance in your checking account was 68. When your bankstatement arrives, you realize that you forgot to record a check. The bankstatement lists your balance as 26. Write and solve an equation to find theamount of the check that you forgot to record.Section 1.1Solving Simple Equations7
Exercises1.1Tutorial Help in English and Spanish at BigIdeasMath.comVocabulary and Core Concept Check1. VOCABULARY Which of the operations , , , and are inverses of each other?2. VOCABULARY Are the equations 2x 10 and 5x 25 equivalent? Explain.3. WRITING Which property of equality would you use to solve the equation 14x 56? Explain.4. WHICH ONE DOESN’T BELONG? Which expression does not belong with the other three? Explainyour reasoning.x8 —23 x 4x3— 9x 6 5Monitoring Progress and Modeling with MathematicsIn Exercises 5–14, solve the equation. Justify each step.Check your solution. (See Example 1.)USING TOOLS The sum of the angle measures of a5. x 5 86. m 9 2quadrilateral is 360 . In Exercises 17–20, write andsolve an equation to find the value of x. Use a protractorto check the reasonableness of your answer.7. y 4 38. s 2 117.9. w 3 410. n 6 711. 14 p 1112. 0 4 q13. r ( 8) 10x x 150 100 77 120 48 100 14. t ( 5) 915. MODELING WITH MATHEMATICS A discounted18.19. 76 amusement park ticket costs 12.95 less than theoriginal price p. Write and solve an equation to findthe original price.92 x 20.115 122 85 x 60 In Exercises 21–30, solve the equation. Justify each step.Check your solution. (See Example 2.)16. MODELING WITH MATHEMATICS You and a friendare playing a board game. Your final score x is12 points less than your friend’s final score. Writeand solve an equation to find your final score.ROUND9ROUND10FINALSCOREYour FriendYou8Chapter 1Solving Linear Equations21. 5g 2022. 4q 5223. p 5 324. y 7 125. 8r 6426. x ( 2) 8x6w 327. — 828. — 629. 54 9s30. 7 —t7
In Exercises 31– 38, solve the equation. Check yoursolution.31.3—2 t 1—232. b 3—16 45. REASONING Identify the property of equality thatmakes Equation 1 and Equation 2 equivalent.5—1633. —37 m 634. —5 y 4Equation 11 xx — — 32 435. 5.2 a 0.436. f 3π 7πEquation 24x 2 x 1237. 108π 6π j38. x ( 2) 1.42ERROR ANALYSIS In Exercises 39 and 40, describe andcorrect the error in solving the equation.39. 46. PROBLEM SOLVING Tatami mats are used as a floorcovering in Japan. One possible layout uses fouridentical rectangular mats and one square mat, asshown. The area of the square mat is half the area ofone of the rectangular mats. 0.8 r 12.6r 12.6 ( 0.8)r 11.840. 3Total area 81 ft2m — 43m — 3 ( 4)3m 12 ( ) 41. ANALYZING RELATIONSHIPS A baker orders 162 eggs.Each carton contains 18 eggs. Which equation canyou use to find the number x of cartons? Explain yourreasoning and solve the equation.A 162x 18 xB — 162 18C 18x 162 D x 18 162 MODELING WITH MATHEMATICS In Exercises 42– 44,write and solve an equation to answer the question.(See Examples 3 and 4.)42. The temperature at 5 p.m. is 20 F. The temperatureat 10 p.m. is 5 F. How many degrees did thetemperature fall?47. PROBLEM SOLVING You spend 30.40 on 4 CDs.Each CD costs the same amount and is on sale for80% of the original price.a. Write and solve anequation to find howmuch you spend oneach CD.b. The next day, the CDsare no longer on sale.You have 25. Will yoube able to buy 3 more CDs?Explain your reasoning.48. ANALYZING RELATIONSHIPS As c increases, doesthe value of x increase, decrease, or stay the samefor each equation? Assume c is positive.43. The length of anAmerican flag is1.9 times its width.What is the width ofthe flag?a. Write and solve an equation to find the area ofone rectangular mat.b. The length of a rectangular mat is twice thewidth. Use Guess, Check, and Revise to findthe dimensions of one rectangular mat.9.5 ftEquationValue of xx c 044. The balance of an investment account is 308 morecx 1than the balance 4 years ago. The current balanceof the account is 4708. What was the balance4 years ago?cx cxc— 1Section 1.1Solving Simple Equations9
49. USING STRUCTURE Use the values 2, 5, 9, and 10MATHEMATICAL CONNECTIONS In Exercises 53–56, findthe height h or the area of the base B of the solid.to complete each statement about the equationax b 5.53.54.a. When a and b , x is a positive integer.hb. When a and b , x is a negative integer.7 in.B 147 cm2B50. HOW DO YOU SEE IT? The circle graph shows theVolume 84π in.3percents of different animals sold at a local pet storein 1 year.55.Hamster: 5%Volume 1323 cm356.5mhRabbit:9%Bird:7%B 30 ft 2BDog:48%Volume 15π m3Cat:x%Volume 35 ft357. MAKING AN ARGUMENT In baseball, a player’sbatting average is calculated by dividing the numberof hits by the number of at-bats. The table showsPlayer A’s batting average and number of at-bats forthree regular seasons.a. What percent is represented by the entire circle?b. How does the equation 7 9 5 48 x 100relate to the circle graph? How can you use thisequation to find the percent of cats sold?SeasonBatting averageAt-bats2010.3125962011.2964462012.29559951. REASONING One-sixth of the girls and two-seventhsof the boys in a school marching band are in thepercussion section. The percussion section has 6 girlsand 10 boys. How many students are in the marchingband? Explain.a. How many hits did Player A have in the 2011regular season? Round your answer to the nearestwhole number.b. Player B had 33 fewer hits in the 2011 season thanPlayer A but had a greater batting average. Yourfriend concludes that Player B had more at-bats inthe 2011 season than Player A. Is your friendcorrect? Explain.52. THOUGHT PROVOKING Write a real-life problemthat can be modeled by an equation equivalent to theequation 5x 30. Then solve the equation and writethe answer in the context of your real-life problem.Maintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsUse the Distributive Property to simplify the expression. (Skills Review Handbook)(159. —56 x —2 458. 8(y 3))60. 5(m 3 n)61. 4(2p 4q 6)Copy and complete the statement. Round to the nearest hundredth, if necessary.(Skills Review Handbook)5LminLh63. — —7 galminqtsec65. — —62. — —64. — —10Chapter 1Solving Linear Equations68 mihmisec8 kmminhmi
1.2TEXAS ESSENTIALKNOWLEDGE AND SKILLSSolving Multi-Step EquationsEssential QuestionHow can you use multi-step equations to solvereal-life problems?Solving for the Angle Measures of a PolygonA.5.AA.10.DWork with a partner. The sum S of the angle measures of a polygon with n sides canbe found using the formula S 180(n 2). Write and solve an equation to find eachvalue of x. Justify the steps in your solution. Then find the angle measures of eachpolygon. How can you check the reasonableness of your answers?a.b.c.50 (30 x) (2x 30) (x 10) 9x JUSTIFYING THESOLUTIONTo be proficient in math,you need to be sure youranswers make sense inthe context of theproblem. For instance,if you find the anglemeasures of a triangle,and they have a sum thatis not equal to 180 , thenyou should check yourwork for mistakes.30 (2x 20) (x 20) 50 x d.(x 17) e.(x 35) (5x 2) (3x 5) (2x 8) f.(3x 16) (8x 8) (5x 10) (x 42) x (4x 18) (3x 7) (4x 15) (2x 25) Writing a Multi-Step EquationWork with a partner.a. Draw an irregular polygon.b. Measure the angles of the polygon. Record the measurements ona separate sheet of paper.c. Choose a value for x. Then, using this value, work backward to assign avariable expression to each angle measure, as in Exploration 1.d. Trade polygons with your partner.e. Solve an equation to find the angle measures of the polygon your partnerdrew. Do your answers seem reasonable? Explain.Communicate Your Answer3. How can you use multi-step equations to solve real-life problems?4. In Exploration 1, you were given the formula for the sum S of the angle measuresof a polygon with n sides. Explain why this formula works.5. The sum of the angle measures of a polygon is 1080º. How many sides does thepolygon have? Explain how you found your answer.Section 1.2Solving Multi-Step Equations11
1.2 LessonWhat You Will LearnSolve multi-step linear equations using inverse operations.Use multi-step linear equations to solve real-life problems.Core VocabulVocabularylarryUse unit analysis to model real-life problems.Previousinverse operationsmeanSolving Multi-Step Linear EquationsCore ConceptSolving Multi-Step EquationsTo solve a multi-step equation, simplify each side of the equation, if necessary.Then use inverse operations to isolate the variable.Solving a Two-Step EquationSolve 2.5x 13 2. Check your solution.SOLUTION2.5x 13 13Undo the subtraction.Write the equation.2 132.5x Add 13 to each side.2.5x15— —2.52.5Undo the multiplication.CheckSimplify.152.5x 13 2?2.5(6) 13 2Divide each side by 2.5.x 62 2Simplify. The solution is x 6.Combining Like Terms to Solve an EquationSolve 12 9x 6x 15. Check your solution.SOLUTIONUndo the addition.Undo the multiplication. 12 9x 6x 15Write the equation. 12 3x 15Combine like terms. 15Subtract 15 from each side. 15 27 3xSimplify. 27 3x— —33Divide each side by 3. 9 xCheckSimplify. 12 12The solution is x 9.Monitoring Progress 12 9x 6x 15? 12 9( 9) 6( 9) 15 Help in English and Spanish at BigIdeasMath.comSolve the equation. Check your solution.1. 2n 3 912Chapter 1Solving Linear Equations2. 21 —12 c 113. 2x 10x 12 18
Using Structure to Solve a Multi-Step EquationSolve 2(1 x) 3 8. Check your solution.SOLUTIONMethod 1 One way to solve the equation is by using the Distributive Property.2(1 x) 3 82(1) 2(x) 3 82 2x 3 8 2x 5 8 5 5Write the equation.Distributive PropertyMultiply.Combine like terms.Subtract 5 from each side. 2x 13Simplify. 2x 2Divide each side by 2. 13 2— —x 6.5The solution is x 6.5.Simplify.Check2(1 x) 3 8?2(1 6.5) 3 8 8 8 Method 2 Another way to solve the equation is by interpreting the expression1 x as a single quantity.ANALYZINGMATHEMATICALRELATIONSHIPSFirst solve for theexpression 1 x, andthen solve for x.2(1 x) 3 8 3Write the equation. 3Subtract 3 from each side.2(1 x) 11Simplify.2(1 x)2Divide each side by 2. 112— —1 x 5.5 1 1Simplify.Subtract 1 from each side. x 6.5Simplify. x 1Divide each side by 1. 6.5 1— —x 6.5Simplify.The solution is x 6.5, which is the same solution obtained in Method 1.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSolve the equation. Check your solution.4. 3(x 1) 6 95. 15 5 4(2d 3)6. 13 2(y 4) 3y7. 2x(5 3) 3x 58. 4(2m 5) 3m 359. 5(3 x) 2(3 x) 14Section 1.2Solving Multi-Step Equations13
Solving Real-Life ProblemsModeling with MathematicsU the table to find the number of miles xUseyyou need to bike on Friday so that the meannnumber of miles biked per day is ursday5Fridayx11. Understand the Problem You know howmany miles you biked Monday throughThursday. You are asked to find the numberof miles you need to bike on Friday so thatthe mean number of miles biked per day is 5.22. Make a Plan Use the definition of mean to write an equation that represents theproblem. Then solve the equation.33. Solve the Problem The mean of a data set is the sum of the data divided by thenumber of data values.3.5 5.5 0 5 x5Write the equation.14 x5Combine like terms.—— 5— 514 x5 — 5 55 14 x 14 Multiply each side by 5.25Simplify. 14Subtract 14 from each side.x 11Simplify.You need to bike 11 miles on Friday.4. Look Back Notice that on the days that you did bike, the values are close tothe mean. Because you did not bike on Wednesday, you need to bike abouttwice the mean on Friday. Eleven miles is about twice the mean. So, yoursolution is reasonable.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com110. The formula d —2 n 26 relates the nozzle pressure n (in pounds per squareinch) of a fire hose and the maximum horizontal distance the water reaches d(in feet). How much pressure is needed to reach a fire 50 feet away?d14Chapter 1Solving Linear Equations
REMEMBERWhen you add miles tomiles, you get miles.But, when you dividemiles by days, youget miles per day.Using Unit Analysis to Model Real-Life ProblemsWhen you write an equation to model a real-life problem, you should check that theunits on each side of the equation balance. For instance, in Example 4, notice howthe units balance.milesmiles per day3.5 5.5 0 5 x5permidaymiday—— 5— — daysSolving a Real-Life ProblemYour school’s drama club charges 4 per person for admission to a play. The clubborrowed 400 to pay for costumes and props. After paying back the loan, the clubhas a profit of 100. How many people attended the play?SOLUTION1. Understand the Problem You know how much the club charges for admission.You also know how much the club borrowed and its profit. You are asked to findhow many people attended the play.2. Make a Plan Use a verbal model to write an equation that represents the problem.Then solve the equation.3. Solve the ProblemREMEMBERWhen you multiply dollarsper person by people, youget dollars. who attendedWordsTicketpriceVariableLet x be the number of people who attended.Equation— 4personNumber of people Amount Profitof loan x people 400 1004x 400 100 Write the equation.4x 400 400 100 4004x 5004x4 Add 400 to each side.Simplify.5004— —Divide each side by 4.x 125Simplify.So, 125 people attended the play.4. Look Back To check that your solution is reasonable, multiply 4 per person by125 people. The result is 500. After paying back the 400 loan, the club has 100,which is the profit.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com11. You have 96 feet of fencing to enclose a rectangular pen for your dog. To providesufficient running space for your dog to exercise, the pen should be three times aslong as it is wide. Find the dimensions of the pen.Section 1.2Solving Multi-Step Equations15
1.2ExercisesTutorial Help in English and Spanish at BigIdeasMath.comVocabulary and Core Concept Check1. COMPLETE THE SENTENCE To solve the equation 2x 3x 20, first combine 2x and 3x becausethey are .2. WRITING Describe two ways to solve the equation 2(4x 11) 10.Monitoring Progress and Modeling with MathematicsIn Exercises 3 14, solve the equation. Check yoursolution. (See Examples 1 and 2.)23. 3(3 x)
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
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
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
Module 4: Linear Equations (40 days) Unit 3: Systems of Linear Equations This unit extends students' facility with solving problems by writing and solving equations. Big Idea: The solution to a system of two linear equations in two variables is an ordered pair that satisfies both equations.
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
15th AMC ! 8 1999 5 Problems 17, 18, and 19 refer to the following: Cookies For a Crowd At Central Middle School the 108 students who take the AMC! 8 meet in the evening to talk about prob-lems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie’s Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, list these items: 11 2 cups of our, 2 eggs .
