5Solving Systems Of Linear Equations
Solving Systems of5 Linear Equations5.15.25.35.45.55.65.7Solving Systems of Linear Equations by GraphingSolving Systems of Linear Equations by SubstitutionSolving Systems of Linear Equations by EliminationSolving Special Systems of Linear EquationsSolving Equations by GraphingGraphing Linear Inequalities in Two VariablesSystemsyof Linear InequalitiesqSEE the Big IdeaFishingFishihing ((p.p. 27279)9)Pets (p. 266)DeliveryDelilivery VansVans (p.(p. 250)250)Drama Club (p(p. 244)Roofing Contractor (p. 238)hsnb alg1 pe 05op.indd 2322/3/16 10:29 AM
Maintaining Mathematical ProficiencyGraphing Linear Functions1Example 1 Graph 3 y — x.2Step 1 Rewrite the equation in slope-intercept form.1y —x 322yStep 2 Find the slope and the y-intercept.x 41m — and b 32 22 1(0, 3)Step 3 The y-intercept is 3. So, plot (0, 3). 4Step 4 Use the slope to find another point on the line.412rise 1slope — —run 2Plot the point that is 2 units right and 1 unit up from (0, 3). Draw a linethrough the two points.Graph the equation.1. y 4 x2. 6x y 13. 4x 5y 204. 2y 12 3xSolving and Graphing Linear InequalitiesExample 2 Solve 2x 17 8x 5. Graph the solution.2x 17 8x 5 5 52x 12 2x8x 2x 12 1266x6x6— —Write the inequality.Add 5 to each side.Simplify.Subtract 2x from each side.Simplify.Divide each side by 6. 2 xSimplify.x –2The solution is x 2. 5 4 3 2 10123Solve the inequality. Graph the solution.5. m 4 96. 24 6t8. 5z 1 149. 4k 16 k 27. 2a 5 1310. 7w 12 2w 311. ABSTRACT REASONING The graphs of the linear functions g and h have different slopes. Thevalue of both functions at x a is b. When g and h are graphed in the same coordinate plane,what happens at the point (a, b)?Dynamic Solutions available at BigIdeasMath.comhsnb alg1 pe 05op.indd 2332332/4/15 4:22 PM
MathematicalPracticesMathematically proficient students use technological tools toexplore concepts.Using a Graphing CalculatorCore ConceptFinding the Point of IntersectionYou can use a graphing calculator to find the point of intersection, if it exists, ofthe graphs of two linear equations.1.Enter the equations into a graphing calculator.2.Graph the equations in an appropriate viewing window, so that the point ofintersection is visible.3.Use the intersect feature of the graphing calculator to find the point ofintersection.Using a Graphing CalculatorUse a graphing calculator to find the point of intersection, if it exists, of the graphs ofthe two linear equations.1y —2 x 2Equation 1y 3x 5Equation 2SOLUTION4The slopes of the lines are not the same, soyou know that the lines intersect. Enter theequations into a graphing calculator. Thengraph the equations in an appropriateviewing window.1y 2 x 2 66y 3x 5 44Use the intersect feature to find the pointof intersection of the lines. 6The point of intersection is (2, 1).6IntersectionX 2Y 1 4Monitoring ProgressUse a graphing calculator to find the point of intersection of the graphs of thetwo linear equations.1. y 2x 3y 2341—2 x 3Chapter 5hsnb alg1 pe 05op.indd 2342. y x 13. 3x 2y 2y x 22x y 2Solving Systems of Linear Equations2/4/15 4:22 PM
5.1Solving Systems of LinearEquations by GraphingEssential QuestionHow can you solve a system of linearequations?Writing a System of Linear EquationsWork with a partner. Your family opens a bed-and-breakfast. They spend 600preparing a bedroom to rent. The cost to your family for food and utilities is 15 per night. They charge 75 per night to rent the bedroom.a. Write an equation that represents the costs. 15 perCost, C night(in dollars) nights, xNumber of 600b. Write an equation that represents the revenue (income). 75 perRevenue, R night(in dollars)MODELING WITHMATHEMATICSTo be proficient in math,you need to identifyimportant quantities inreal-life problems andmap their relationshipsusing tools such asdiagrams, tables,and graphs. nights, xNumber ofc. A set of two (or more) linear equations is called a system of linear equations.Write the system of linear equations for this problem.Using a Table or Graph to Solve a SystemWork with a partner. Use the cost and revenue equations from Exploration 1 todetermine how many nights your family needs to rent the bedroom before recoveringthe cost of preparing the bedroom. This is the break-even point.a. Copy and complete the table.x (nights)01234567891011C (dollars)R (dollars)b. How many nights does your family need to rent the bedroom before breaking even?c. In the same coordinate plane, graph the cost equation and the revenue equationfrom Exploration 1.d. Find the point of intersection of the two graphs. What does this point represent?How does this compare to the break-even point in part (b)? Explain.Communicate Your Answer3. How can you solve a system of linear equations? How can you check yoursolution?4. Solve each system by using a table or sketching a graph. Explain why you choseeach method. Use a graphing calculator to check each solution.a. y 4.3x 1.3y 1.7x 4.7Section 5.1hsnb alg1 pe 0501.indd 235b. y xy 3x 8c. y x 1y 3x 5Solving Systems of Linear Equations by Graphing2352/4/15 4:24 PM
5.1LessonWhat You Will LearnCheck solutions of systems of linear equations.Solve systems of linear equations by graphing.Core VocabulVocabularylarryUse systems of linear equations to solve real-life problems.system of linear equations,p. 236solution of a system of linearequations, p. 236Previouslinear equationordered pairSystems of Linear EquationsA system of linear equations is a set of two or more linear equations in the samevariables. An example is shown below.x y 7Equation 12x 3y 11Equation 2A solution of a system of linear equations in two variables is an ordered pair that is asolution of each equation in the system.Checking SolutionsTell whether the ordered pair is a solution of the system of linear equations.a. (2, 5); x y 72x 3y 11b. ( 2, 0); y 2x 4y x 4Equation 1Equation 2Equation 1Equation 2SOLUTIONa. Substitute 2 for x and 5 for y in each equation.Equation 1READINGA system of linearequations is also calleda linear system.Equation 2x y 72x 3y 11?2 5 72(2) 3(5) 117 7? 11 11 Because the ordered pair (2, 5) is a solution of each equation, it is a solution ofthe linear system.b. Substitute 2 for x and 0 for y in each equation.Equation 1Equation 2y 2x 4y x 4?0 2( 2) 40 2 40 0 ?0 2 The ordered pair ( 2, 0) is a solution of the first equation, but it is not a solutionof the second equation. So, ( 2, 0) is not a solution of the linear system.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comTell whether the ordered pair is a solution of the system of linear equations.1. (1, 2);236Chapter 5hsnb alg1 pe 0501.indd 2362x y 0 x 2y 52. (1, 4);y 3x 1y x 5Solving Systems of Linear Equations2/4/15 4:24 PM
Solving Systems of Linear Equations by GraphingThe solution of a system of linear equations is the point of intersection of the graphs ofthe equations.Core ConceptSolving a System of Linear Equations by GraphingREMEMBERNote that the linearequations are inslope-intercept form. Youcan use the methodpresented in Section 3.5to graph the equations.Step 1Graph each equation in the same coordinate plane.Step 2Estimate the point of intersection.Step 3Check the point from Step 2 by substituting for x and y in each equationof the original system.Solving a System of Linear Equations by GraphingSolve the system of linear equations by graphing.y 2x 5Equation 1y 4x 1Equation 2SOLUTIONyStep 1 Graph each equation.Step 2 Estimate the point of intersection.The graphs appear to intersect at (1, 3).(1,1 3)Step 3 Check your point from Step 2.2Equation 1Equation 2y 2x 5y 4x 1?3 2(1) 5?3 4(1) 13 33 3 y 4x 1y 2x 5 4 2 124 x The solution is (1, 3).Check6y 2x 5 6IntersectionX 1Y 3 2Monitoring Progressy 4x 16Help in English and Spanish at BigIdeasMath.comSolve the system of linear equations by graphing.3. y x 2y x 4Section 5.1hsnb alg1 pe 0501.indd 23714. y —2 x 3y 3 —2 x 55. 2x y 53x 2y 4Solving Systems of Linear Equations by Graphing2372/4/15 4:24 PM
Solving Real-Life ProblemsModeling with MathematicsA roofing contractor buys 30 bundles of shingles and 4 rolls of roofing paper for 1040. In a second purchase (at the same prices), the contractor buys 8 bundles ofsshingles for 256. Find the price per bundle of shingles and the price per roll ofrroofing paper.SOLUTIONS11. Understand the Problem You know the total price of each purchase and howmany of each item were purchased. You are asked to find the price of each item.22. Make a Plan Use a verbal model to write a system of linear equations thatrepresents the problem. Then solve the system of linear equations.33. Solve the ProblemWords308per Pricebundle bundlePrice per 4 0 Priceper roll Priceper roll 1040 256Variables Let x be the price (in dollars) per bundle and let y be theprice (in dollars) per roll.System30x 4y 1040Equation 18x 256Equation 2Step 1 Graph each equation. Note that onlythe first quadrant is shown becausex and y must be positive.Step 2 Estimate the point of intersection. Thegraphs appear to intersect at (32, 20).Step 3 Check your point from Step 2.Equation 130x 4y 1040?30(32) 4(20) 10401040 1040y320y 7.5x 260240x 3216080Equation 2 8x 256?8(32) 256256 2560(32, 20)08162432x The solution is (32, 20). So, the price per bundle of shingles is 32, and theprice per roll of roofing paper is 20.4. Look Back You can use estimation to check that your solution is reasonable.A bundle of shingles costs about 30. So, 30 bundles of shingles and 4 rolls ofroofing paper (at 20 per roll) cost about 30(30) 4(20) 980, and 8 bundlesof shingles costs about 8(30) 240. These prices are close to the given values,so the solution seems reasonable.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com6. You have a total of 18 math and science exercises for homework. You havesix more math exercises than science exercises. How many exercises do youhave in each subject?238Chapter 5hsnb alg1 pe 0501.indd 238Solving Systems of Linear Equations2/4/15 4:24 PM
5.1ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. VOCABULARY Do the equations 5y 2x 18 and 6x 4y 10 form a system of linearequations? Explain.2. DIFFERENT WORDS, SAME QUESTION Consider the system of linear equations 4x 2y 4and 4x y 6. Which is different? Find “both” answers.Solve the system of linear equations.Solve each equation for y.Find the point of intersectionof the graphs of the equations.Find an ordered pair that is a solutionof each equation in the system.Monitoring Progress and Modeling with MathematicsIn Exercises 3–8, tell whether the ordered pair isa solution of the system of linear equations.(See Example 1.)x y 6x y 83. (2, 6);4. (8, 2);2x 10y 43x y 05. ( 1, 3);7. ( 2, 1);13. y x 718. 4x 4y 2019. x 4y 420. 3y 4x 32x y 4y 2x 4yyx 2 3x 4y 1221.42 x111. 6y 3x 1822.y4x2 yThe solution ofthe linear systemx 3y 6 and2x 3y 3is (3, 1).The solution ofthe linear systemy 2x 1 andy x 1is x 2.422 22xxSection 5.1hsnb alg1 pe 0501.indd 239y 12x 4y 84 224y 4x 3y 612. 2x y 2 x 4y 24 6y 5ERROR ANALYSIS In Exercises 21 and 22, describeand correct the error in solving the system of linearequations.10. x y 541y —2 x 1117. 9x 3y 3In Exercises 9–12, use the graph to solve the system oflinear equations. Check your solution.2316. y —4 x 4y —23 x 56x 5y 76x 3y 128. (5, 6);2x 4y 84x y 144x y 1y 2x 815. y —3 x 2y 2x 6y 3x 149. x y 414. y x 4y x 11y 7x 4y 8x 56. ( 4, 2);In Exercises 13–20, solve the system of linear equationsby graphing. (See Example 2.)24xSolving Systems of Linear Equations by Graphing2392/4/15 4:24 PM
USING TOOLS In Exercises 23–26, use a graphing31. COMPARING METHODS Consider the equationcalculator to solve the system of linear equations.x 2 3x 4.23. 0.2x 0.4y 4a. Solve the equation using algebra.24. 1.6x 3.2y 24 0.6x 0.6y 32.6x 2.6y 2625. 7x 6y 0b. Solve the system of linear equations y x 2and y 3x 4 by graphing.26. 4x y 1.50.5x y 2c. How is the linear system and the solution in part(b) related to the original equation and the solutionin part (a)?2x y 1.527. MODELING WITH MATHEMATICS You have40 minutes to exercise at the gym, and you want toburn 300 calories total using both machines. Howmuch time should you spend on each machine?(See Example 3.)Elliptical Trainer32. HOW DO YOU SEE IT? A teacher is purchasingbinders for students. The graph shows the total costsof ordering x binders from three different companies.Stationary BikeCost (dollars)Buying Bindersy1501256 caloriesper minuteCompany C75500 15 20 25 30 35 40 45 50 xNumber of binders28. MODELING WITH MATHEMATICSYou sell small and large candlesat a craft fair. You collect 144selling a total of 28 candles.How many of each type of candledid you sell?Company A10008 caloriesper minuteCompany Ba. For what numbers of binders are the costs thesame at two different companies? Explain. 6eachb. How do your answers in part (a) relate to systemsof linear equations? 4each33. MAKING AN ARGUMENT You and a friend are going29. MATHEMATICAL CONNECTIONS Write a linearhiking but start at different locations. You start at thetrailhead and walk 5 miles per hour. Your friend starts3 miles from the trailhead and walks 3 miles per hour.equation that represents the area and a linear equationthat represents the perimeter of the rectangle. Solvethe system of linear equations by graphing. Interpretyour solution.you(3x 3) cm6 cmyour friend30. THOUGHT PROVOKING Your friend’s bank accountbalance (in dollars) is represented by the equationy 25x 250, where x is the number of months.Graph this equation. After 6 months, you want tohave the same account balance as your friend. Write alinear equation that represents your account balance.Interpret the slope and y-intercept of the line thatrepresents your account balance.Maintaining Mathematical ProficiencySolve the literal equation for y.34. 10x 5y 5x 20240Chapter 5hsnb alg1 pe 0501.indd 240a. Write and graph a system of linear equations thatrepresents this situation.b. Your friend says that after an hour of hiking youwill both be at the same location on the trail. Isyour friend correct? Use the graph from part (a) toexplain your answer.Reviewing what you learned in previous grades and lessons(Section 1.5)35. 9x 18 6y 3x136. —34 x —4 y 5Solving Systems of Linear Equations2/4/15 4:24 PM
5.2Solving Systems of LinearEquations by SubstitutionEssential QuestionHow can you use substitution to solve a systemof linear equations?Using Substitution to Solve SystemsWork with a partner. Solve each system of linear equations using two methods.Method 1 Solve for x first.Solve for x in one of the equations. Substitute the expression for x into the otherequation to find y. Then substitute the value of y into one of the original equationsto find x.Method 2 Solve for y first.Solve for y in one of the equations. Substitute the expression for y into the otherequation to find x. Then substitute the value of x into one of the original equationsto find y.Is the solution the same using both methods? Explain which method you would preferto use for each system.a. x y 7b. x 6y 11 5x y 53x 2y 7c. 4x y 13x 5y 18Writing and Solving a System of EquationsWork with a partner.a. Write a random ordered pair with integercoordinates. One way to do this is to usea graphing calculator. The ordered pairgenerated at the right is ( 2, 3).ATTENDING TOPRECISIONTo be proficient in math,you need to communicateprecisely with others.b. Write a system of linear equations that hasyour ordered pair as its solution.Choose tworandom integersbetween 5 and 5.randInt(-5‚5‚2){-2 -3}c. Exchange systems with your partner and useone of the methods from Exploration 1 tosolve the system. Explain your choiceof method.Communicate Your Answer3. How can you use substitution to solve a system of linear equations?4. Use one of the methods from Exploration 1 to solve each system of linearequations. Explain your choice of method. Check your solutions.a. x 2y 7b. x 2y 62x y 92x y 2d. 3x 2y 13x 3y 3Section 5.2hsnb alg1 pe 0502.indd 241e. 3x 2y 9 x 3y 8c. 3x 2y 10 2x y 6f. 3x y 64x 5y 11Solving Systems of Linear Equations by Substitution2412/4/15 4:24 PM
5.2 LessonWhat You Will LearnSolve systems of linear equations by substitution.Use systems of linear equations to solve real-life problems.Core VocabulVocabularylarryPrevioussystem of linear equationssolution of a system oflinear equationsSolving Linear Systems by SubstitutionAnother way to solve a system of linear equations is to use substitution.Core ConceptSolving a System of Linear Equations by SubstitutionStep 1 Solve one of the equations for one of the variables.Step 2 Substitute the expression from Step 1 into the other equation andsolve for the other variable.Step 3 Substitute the value from Step 2 into one of the original equationsand solve.Solving a System of Linear Equationsby SubstitutionSolve the system of linear equations by substitution.y 2x 9Equation 16x 5y 19Equation 2SOLUTIONStep 1 Equation 1 is already solved for y.Step 2 Substitute 2x 9 for y in Equation 2 and solve for x.6x 5y 19CheckEquation 26x 5( 2x 9) 19Equation 1y 2x 9? 1 2( 4) 9 1 1Substitute 2x 9 for y.6x 10x 45 19Distributive Property16x 45 19Combine like terms.16x 64 Subtract 45 from each side.x 4Divide each side by 16.Step 3 Substitute 4 for x in Equation 1 and solve for y.Equation 2y 2x 96x 5y 19?6( 4) 5( 1) 19 19 19 Equation 1 2( 4) 9Substitute 4 for x. 8 9Multiply. 1Subtract.The solution is ( 4, 1).Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSolve the system of linear equations by substitution. Check your solution.1. y 3x 14y 4x242Chapter 5hsnb alg1 pe 0502.indd 2422. 3x 2y 0y 1—2 x 13. x 6y 74x y 3Solving Systems of Linear Equations2/4/15 4:24 PM
Solving a System of Linear Equationsby SubstitutionANOTHER WAYYou could also begin bysolving for x in Equation 1,solving for y in Equation 2,or solving for x inEquation 2.Solve the system of linear equations by substitution. x y 3Equation 13x y 1Equation 2SOLUTIONStep 1 Solve for y in Equation 1.y x 3Revised Equation 1Step 2 Substitute x 3 for y in Equation 2 and solve for x.3x y 13x (x 3) 14x 3 14x 4x 1Equation 2Substitute x 3 for y.Combine like terms.Subtract 3 from each side.Divide each side by 4.Step 3 Substitute 1 for x in Equation 1 and solve for y. x y 3 ( 1) y 3y 2Equation 1Substitute 1 for x.Subtract 1 from each side.The solution is ( 1, 2).Graphical CheckAlgebraic Check4Equation 1 x y 3? ( 1) 2 33 3y x 3 Equation 2y 3x 1 54IntersectionX -1Y 2 23x y 1?3( 1) 2 1 1 1 Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSolve the system of linear equations by substitution. Check your solution.4. x y 25. x y 4 3x y 66. 2x y 53x y 1Section 5.2hsnb alg1 pe 0502.indd 2434x y 107. x 2y 73x 2y 3Solving Systems of Linear Equations by Substitution2432/4/15 4:24 PM
Solving Real-Life ProblemsModeling with MathematicsA drama club earns 1040 from a production. A total of 64 adult tickets and132 student tickets are sold. An adult ticket costs twice as much as a student ticket.Write a system of linear equations that represents this situation. What is the priceWoof each type of ticket?SOLUTIONS11. Understand the Problem You know the amount earned, the total numbers of adultand student tickets sold, and the relationship between the price of an adult ticketand the price of a student ticket. You are asked to write a system of linear equationsthat represents the situation and find the price of each type of ticket.22. Make a Plan Use a verbal model to write a system of linear equations thatrepresents the problem. Then solve the system of linear equations.33. Solve the ProblemWords64ticket AdultpriceAdult ticket 2price 132 Studentticket price 1040 Studentticket priceVariables Let x be the price (in dollars) of an adult ticket and let y be theprice (in dollars) of a student ticket.System64x 132y 1040Equation 1x 2yEquation 2Step 1 Equation 2 is already solved for x.Step 2 Substitute 2y for x in Equation 1 and solve for y.64x 132y 1040STUDY TIPYou can use either ofthe original equationsto solve for x. However,using Equation 2 requiresfewer calculations.Equation 164(2y) 132y 1040Substitute 2y for x.260y 1040Simplify.y 4Simplify.Step 3 Substitute 4 for y in Equation 2 and solve for x.x 2yEquation 2x 2(4)Substitute 4 for y.x 8Simplify.The solution is (8, 4). So, an adult ticket costs 8 and a student ticket costs 4.4. Look Back To check that your solution is correct, substitute the values of x andy into both of the original equations and simplify.64(8) 132(4) 10401040 1040Monitoring Progress 8 2(4)8 8 Help in English and Spanish at BigIdeasMath.com8. There are a total of 64 students in a drama club and a yearbook club. The dramaclub has 10 more students than the yearbook club. Write a system of linearequations that represents this situation. How many students are in each club?244Chapter 5hsnb alg1 pe 0502.indd 244Solving Systems of Linear Equations2/4/15 4:24 PM
5.2ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. WRITING Describe how to solve a system of linear equations by substitution.2. NUMBER SENSE When solving a system of linear equations by substitution, how do you decidewhich variable to solve for in Step 1?Monitoring Progress and Modeling with MathematicsIn Exercises 3 8, tell which equation you would chooseto solve for one of the variables. Explain.3. x 4y 30x 2y 04. 3x y 05x y 56. 3x 2y 194x 3y 58. 3x 5y 25x 2y 6In Exercises 9–16, solve the sytem of linear equationsby substitution. Check your solution. (See Examples 1and 2.)9. x 17 4yy x 23x 4y 813. 2x 1215. 5x 2y 9x y 3corn and wheat on a 180-acre farm. The farmer wantsto plant three times as many acres of corn as wheat.Write a system of linear equations that represents thissituation. How many acres of each crop should thefarmer plant? (See Example 3.)12. 5x 3y 51y 10x 8x 9 120. MODELING WITH MATHEMATICS A company that16. 11x 7y 14x 2y 417. ERROR ANALYSIS Describe and correct the error insolving for one of the variables in the linear system8x 2y 12 and 5x y 4. 3x y 93x 6 93x 3x 119. MODELING WITH MATHEMATICS A farmer plants14. 2x y 23x 5y 29Step 310. 6x 9 yy 3x11. x 16 4yStep 1 3x y 9y 9 3xStep 2 4x 2(9 3x) 64x 18 6x 6 2x 12x 6x y 87. x y 3solving for one of the variables in the linear system4x 2y 6 and 3x y 9. 2x y 105. 5x 3y 1118. ERROR ANALYSIS Describe and correct the error inoffers tubing trips down a river rents tubes for aperson to use and “cooler” tubes to carry food andwater. A group spends 270 to rent a total of 15 tubes.Write a system of linear equations that represents thissituation. How many of each type of tube does thegroup rent?Step 1 5x y 4 y 5x 4y 5x 4Step 2 5x (5x 4) 45x 5x 4 44 4Section 5.2hsnb alg1 pe 0502.indd 245Solving Systems of Linear Equations by Substitution2452/4/15 4:25 PM
30. MAKING AN ARGUMENT Your friend says that givenIn Exercises 21–24, write a system of linear equationsthat has the ordered pair as its solution.21. (3, 5)22. ( 2, 8)23. ( 4, 12)24. (15, 25)a linear system with an equation of a horizontal lineand an equation of a vertical line, you cannot solvethe system by substitution. Is your friend correct?Explain.31. OPEN-ENDED Write a system of linear equations in25. PROBLEM SOLVING A math test is worth 100 pointswhich (3, 5) is a solution of Equation 1 but not asolution of Equation 2, and ( 1, 7) is a solution ofthe system.and has 38 problems. Each problem is worth either5 points or 2 points. How many problems of eachpoint value are on the test?32. HOW DO YOU SEE IT? The graphs of two linear26. PROBLEM SOLVING An investor owns shares ofequations are shown.Stock A and Stock B. The investor owns a total of200 shares with a total value of 4000. How manyshares of each stock does the investor own?yy x 16StockPriceA 9.50B 27.0041MATHEMATICAL CONNECTIONS In Exercises 27 and 28,2(a) write an equation that represents the sum of theangle measures of the triangle and (b) use your equationand the equation shown to find the values of x and y.27.y 6 4x246xa. At what point do the lines appear to intersect?b. Could you solve a system of linear equations bysubstitution to check your answer in part (a)?Explain.x x 2 3yy 28.33. REPEATED REASONING A radio station plays a totalof 272 pop, rock, and hip-hop songs during a day. Thenumber of pop songs is 3 times the number of rocksongs. The number of hip-hop songs is 32 more thanthe number of rock songs. How many of each type ofsong does the radio station play?y (y 18) 3x 5y 22x 34. THOUGHT PROVOKING You have 2.65 in coins.Write a system of equations that represents thissituation. Use variables to represent the number ofeach type of coin.29. REASONING Find the values of a and b so that thesolution of the linear system is ( 9, 1).ax by 31ax by 41Equation 1Equation 235. NUMBER SENSE The sum of the digits of atwo-digit number is 11. When the digits are reversed,the number increases by 27. Find the original number.Maintaining Mathematical ProficiencyFind the sum or difference.Reviewing what you learned in previous grades and lessons(Skills Review Handbook)36. (x 4) (2x 7)37. (5y 12) ( 5y 1)38. (t 8) (t 15)39. (6d 2) (3d 3)40. 4(m 2) 3(6m 4)41. 2(5v 6) 6( 9v 2)246Chapter 5hsnb alg1 pe 0502.indd 246Solving Systems of Linear Equations2/4/15 4:25 PM
5.3Solving Systems of LinearEquations by EliminationEssential QuestionHow can you use elimination to solve a systemof linear equations?Writing and Solving a System of EquationsWork with a partner. You purchase a drink and a sandwich for 4.50. Your friendpurchases a drink and five sandwiches for 16.50. You want to determine the price ofa drink and the price of a sandwich.a. Let x represent the price (in dollars) of one drink. Let y represent the price(in dollars) of one sandwich. Write a system of equations for the situation. Usethe following verbal model.Numberof drinksNumber ofPrice per Pricesandwiches sandwichper drink TotalpriceLabel one of the equations Equation 1 and the other equation Equation 2.b. Subtract Equation 1 from Equation 2. Explain how you can use the result to solvethe system of equations. Then find and interpret the solution.CHANGING COURSETo be proficient in math,you need to monitor andevaluate your progressand change course usinga different solutionmethod, if necessary.Using Elimination to Solve SystemsWork with a partner. Solve each system of linear equations using two methods.Method 1 Subtract. Subtract Equation 2 from Equation 1. Then use the result tosolve the system.Method 2 Add. Add the two equations. Then use the result to solve the system.Is the solution the same using both methods? Which method do you prefer?a. 3x y 6b. 2x y 63x y 02x y 2c. x 2y 7x 2y 5Using Elimination to Solve a SystemWork with a partner.2x y 7x 5y 17Equation 1Equation 2a. Can you eliminate a variable by adding or subtracting the equations as they are?If not, what do you need to do to one or both equations so that you can?b. Solve the system individually. Then exchange solutions with your partner andcompare and check the solutions.Communicate Your Answer4. How can you use elimination to solve a system of linear equations?5. When can you add or subtract the equations in a system to solve the system?When do you have to multiply first? Justify your answers with examples.6. In Exploration 3, why can you multiply an equation in the system by a constantand not change the solution of the system? Explain your reasoning.Section 5.3hsnb alg1 pe 0503.indd 247Solving Systems of Linear Equations by Elimination2472/4/15 4:25 PM
5.3 LessonWhat You Will LearnSolve systems of linear equations by elimination.Use systems of linear equations to solve real-life problems.Core VocabulVocabularylarrySolving Linear Systems by EliminationPreviouscoefficientCore ConceptSolving a System of Linear Equations by EliminationStep 1 Multiply, if necessary, one or both equations by a constant so at least onepair of like terms has the same or opposite coefficients.Step 2 Add or subtract the equations to eliminate one of the variables.Step 3 Solve the resulting equation.Step 4 Substitute the value from Step 3 into one of the original equations andsolve for the other variable.You can use elimination to solve a system of equations because replacing oneequation in the system with the sum of that equation and a multiple of the otherproduces a system that has the same solution. Here is why.System 1a bc dEquation 1Equation 2System 2a kc b kdc dEquation 3Equation 2Consider System 1. In this system, a and c are algebraic expressions, and b and d areconstants. Begin by multiplying each side of Equation 2 by a constant k. By theMultiplication Property of Equality, kc kd. You can rewrite Equation 1 asEquation 3 by adding kc on the left and kd on the right. You can rewrite Equation 3 asEquation 1 by subtracting kc on the left and kd on the
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
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 .
These two linear equations model the situation: 12s 24l 780 s l 20 These two equations form a system of linear equations in two variables, sand l. A system of linear equations is often referred to as a linear system. A solution of a linear system is a pair of values of s and l that satisfy both equations.
Unit 12: Media Lesson Section 12.1: Systems of Linear Equations Definitions Two linear equations that relate the same two variables are called a system of linear equations. A solution to a system of linear equations is an ordered pair that satisfies both equations. Example 1: Verify that the point (5, 4) is a soluti
Linear and quadratic equations CONTENTS Examples: Solving linear equations 2 Questions on solving linear equations using a CAS calculator . Year 11 Linear and quadratic equations Page 10 of 12 Answers Linear equation questions Quadratic equation questions Equation graphing question
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.
3.1 Theory of Linear Equations 97 HIGHER-ORDER 3 DIFFERENTIAL EQUATIONS 3.1 Theory of Linear Equations 3.1.1 Initial-Value and Boundary-Value Problems 3.1.2 Homogeneous Equations 3.1.3 Nonhomogeneous Equations 3.2 Reduction of Order 3.3 Homogeneous Linear Equations with Constant Coeffi cients 3.4 Undetermined Coeffi cients 3.5 V
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 .
