Solving Linear Systems - Big Ideas Learning
1.4Solving Linear SystemsEssential QuestionHow can you determine the number ofsolutions of a linear system?A linear system is consistent when it has at least one solution. A linear system isinconsistent when it has no solution.Recognizing Graphs of Linear SystemsWork with a partner. Match each linear system with its corresponding graph.Explain your reasoning. Then classify the system as consistent or inconsistent.a. 2x 3y 3 4x 6y 6b. 2x 3y 3x 2y 5c. 2x 3y 3 4x 6y 6A.B.C.yyy222 24x24x2 2 2 2 24x2Solving Systems of Linear EquationsWork with a partner. Solve each linear system by substitution or elimination. Thenuse the graph of the system below to check your solution.a. 2x y 5x y 1b. x 3y 1 x 2y 4yc. x y 03x 2y 14yy2224x 2x2x 2 2FINDING ANENTRY POINTTo be proficient in math,you need to look for entrypoints to the solution ofa problem. 2 4Communicate Your Answer3. How can you determine the number of solutions of a linear system?4. Suppose you were given a system of three linear equations in three variables.Explain how you would approach solving such a system.5. Apply your strategy in Question 4 to solve the linear system.x y z 1Equation 1x y z 3Equation 2 x y z 1Equation 3Section 1.4hsnb alg2 pe 0104.indd 29Solving Linear Systems292/5/15 9:56 AM
1.4 LessonWhat You Will LearnVisualize solutions of systems of linear equations in three variables.Solve systems of linear equations in three variables algebraically.Core VocabulVocabularylarrySolve real-life problems.linear equation in threevariables, p. 30system of three linearequations, p. 30solution of a system of threelinear equations, p. 30ordered triple, p. 30Previoussystem of two linear equationsVisualizing Solutions of SystemsA linear equation in three variables x, y, and z is an equation of the formax by cz d, where a, b, and c are not all zero.The following is an example of a system of three linear equations inthree variables.3x 4y 8z 3Equation 1x y 5z 12Equation 24x 2y z 10Equation 3A solution of such a system is an ordered triple (x, y, z) whose coordinates makeeach equation true.The graph of a linear equation in three variables is a plane in three-dimensionalspace. The graphs of three such equations that form a system are three planes whoseintersection determines the number of solutions of the system, as shown in thediagrams below.Exactly One SolutionThe planes intersect in a single point,which is the solution of the system.Infinitely Many SolutionsThe planes intersect in a line. Everypoint on the line is a solution of the system.The planes could also be the same plane.Every point in the plane is a solutionof the system.No SolutionThere are no points in common with all three planes.30Chapter 1hsnb alg2 pe 0104.indd 30Linear Functions2/5/15 9:56 AM
Solving Systems of Equations AlgebraicallyThe algebraic methods you used to solve systems of linear equations in two variablescan be extended to solve a system of linear equations in three variables.Core ConceptSolving a Three-Variable SystemStep 1 Rewrite the linear system in three variables as a linear system in twovariables by using the substitution or elimination method.Step 2 Solve the new linear system for both of its variables.Step 3 Substitute the values found in Step 2 into one of the original equations andsolve for the remaining variable.When you obtain a false equation, such as 0 1, in any of the steps, the systemhas no solution.When you do not obtain a false equation, but obtain an identity such as 0 0,the system has infinitely many solutions.LOOKING FORSTRUCTUREThe coefficient of 1 inEquation 3 makes y aconvenient variableto eliminate.Solving a Three-Variable System (One Solution)Solve the system.4x 2y 3z 122x 3y 5z 76x y 4z 3Equation 1Equation 2Equation 3SOLUTIONStep 1 Rewrite the system as a linear system in two variables.ANOTHER WAYIn Step 1, you could alsoeliminate x to get twoequations in y and z, oryou could eliminate zto get two equationsin x and y.4x 2y 3z 1212x 2y 8z 616x 11z 6Add 2 times Equation 3 toEquation 1 (to eliminate y).2x 3y 5z 7 18x 3y 12z 9 16x 7z 2Add 3 times Equation 3 toEquation 2 (to eliminate y).New Equation 1New Equation 2Step 2 Solve the new linear system for both of its variables.16x 11z 6 16x 7z 24z 8z 2x 1Add new Equation 1and new Equation 2.Solve for z.Substitute into new Equation 1 or 2 to find x.Step 3 Substitute x 1 and z 2 into an original equation and solve for y.6x y 4z 36( 1) y 4(2) 3y 5Write original Equation 3.Substitute 1 for x and 2 for z.Solve for y.The solution is x 1, y 5, and z 2, or the ordered triple ( 1, 5, 2).Check this solution in each of the original equations.Section 1.4hsnb alg2 pe 0104.indd 31Solving Linear Systems312/5/15 9:57 AM
Solving a Three-Variable System (No Solution)Solve the system.x y z 2Equation 15x 5y 5z 3Equation 24x y 3z 6Equation 3SOLUTIONStep 1 Rewrite the system as a linear system in two variables. 5x 5y 5z 105x 5y 5z 30 7Add 5 times Equation 1to Equation 2.Because you obtain a false equation, the original system has no solution.Solving a Three-Variable System (Many Solutions)ANOTHER WAYSubtracting Equation 2from Equation 1 givesz 0. After substituting0 for z in each equation,you can see that each isequivalent to y x 3.Solve the system.x y z 3Equation 1x y z 3Equation 25x 5y z 15Equation 3SOLUTIONStep 1 Rewrite the system as a linear system in two variables.x y z 3x y z 32x 2y 6Add Equation 1 toEquation 2 (to eliminate z).x y z 35x 5y z 156x 6y 18Add Equation 2 toEquation 3 (to eliminate z).New Equation 2New Equation 3Step 2 Solve the new linear system for both of its variables. 6x 6y 18Add 3 times new Equation 2to new Equation 3.6x 6y 180 0Because you obtain the identity 0 0, the system has infinitelymany solutions.Step 3 Describe the solutions of the system using an ordered triple. One way to dothis is to solve new Equation 2 for y to obtain y x 3. Then substitutex 3 for y in original Equation 1 to obtain z 0.So, any ordered triple of the form (x, x 3, 0) is a solution of the system.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comSolve the system. Check your solution, if possible.1. x 2y z 113x 2y z 7 x 2y 4z 92. x y z 14x 4y 4z 23x 2y z 03. x y z 8x y z 82x y 2z 164. In Example 3, describe the solutions of the system using an ordered triple interms of y.32Chapter 1hsnb alg2 pe 0104.indd 32Linear Functions2/5/15 9:57 AM
Solving Real-Life ProblemsSolving a Multi-Step ProblemLAWNBBBAABBASTAGEAn amphitheater charges 75 for each seat in Section A, 55 for each seat inSection B, and 30 for each lawn seat. There are three times as many seats inSection B as in Section A. The revenue from selling all 23,000 seats is 870,000.How many seats are in each section of the amphitheater?SOLUTIONStep 1 Write a verbal model for the situation.Number of 3seats in B, yof Numberseats in A, xTotal numberNumber ofNumber ofNumber of of seatsseats in A, xlawn seats, zseats in B, y75of Numberseats in A, x 55of Numberseats in B, y 30of Numberlawn seats, z TotalrevenueStep 2 Write a system of equations.y 3xEquation 1x y z 23,000Equation 275x 55y 30z 870,000Equation 3Step 3 Rewrite the system in Step 2 as a linear system in two variables by substituting3x for y in Equations 2 and 3.x y z 23,000x 3x z 23,0004x z 23,00075x 55y 30z 870,00075x 55(3x) 30z 870,000240x 30z 870,000Write Equation 2.Substitute 3x for y.New Equation 2Write Equation 3.Substitute 3x for y.New Equation 3Step 4 Solve the new linear system for both of its variables.STUDY TIPWhen substituting to findvalues of other variables,choose original or newequations that areeasiest to use. 120x 30z 690,000240x 30z 870,000120x 180,000x 1500y 4500z 17,000Add 30 times new Equation 2to new Equation 3.Solve for x.Substitute into Equation 1 to find y.Substitute into Equation 2 to find z.The solution is x 1500, y 4500, and z 17,000, or (1500, 4500, 17,000). So,there are 1500 seats in Section A, 4500 seats in Section B, and 17,000 lawn seats.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com5. WHAT IF? On the first day, 10,000 tickets sold, generating 356,000 in revenue.The number of seats sold in Sections A and B are the same. How many lawn seatsare still available?Section 1.4hsnb alg2 pe 0104.indd 33Solving Linear Systems332/5/15 9:57 AM
1.4ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. VOCABULARY The solution of a system of three linear equations is expressed as a(n) .2. WRITING Explain how you know when a linear system in three variables has infinitelymany solutions.Monitoring Progress and Modeling with MathematicsIn Exercises 3–8, solve the system using the eliminationmethod. (See Example 1.)3. x y 2z 54. x 4y 6z 1 x 2y z 22x 3y z 92x y 2z 7 x 2y 4z 55. 2x y z 96. 3x 2y z 8 x 6y 2z 175x 7y z 47. 2x 2y 5z 12x y z 22x 4y 3z 14 3x 4y 5z 14x 3y 4z 148. 3x 2y 3z 27x 2y 5z 142x 4y z 613. x 3y z 214. x 2y z 3x y z 03x 2y 3z 115. x 2y 3z 4 2x y z 16x 3y z 716. 2x 3y z 6 3x 2y z 12 2x 2y 4z 14x y z 57x 8y 6z 3117. MODELING WITH MATHEMATICS Three orders areplaced at a pizza shop. Two small pizzas, a liter ofsoda, and a salad cost 14; one small pizza, a literof soda, and three salads cost 15; and three smallpizzas, a liter of soda, and two salads cost 22.How much does each item cost?ERROR ANALYSIS In Exercises 9 and 10, describe andcorrect the error in the first step of solving the system oflinear equations.4x y 2z 18 x 2y z 113x 3y 4z 449.10. 4x y 2z 18 4x 2y z 11y 3z 712x 3y 6z 183x 3y 4z 4415x 2z 2618. MODELING WITH MATHEMATICS Sam’s FurnitureStore places the following advertisement in the localnewspaper. Write a system of equations for the threecombinations of furniture. What is the price of eachpiece of furniture? Explain.SAM’SuFurnitre StoreSofa and love seatIn Exercises 11–16, solve the system using theelimination method. (See Examples 2 and 3.)11. 3x y 2z 46x 2y 4z 82x y 3z 1034Chapter 1hsnb alg2 pe 0104.indd 3412. 5x y z 6x y z 212x 4y 10Sofa and two chairsSofa, love seat, and one chairLinear Functions2/5/15 9:57 AM
In Exercises 19–28, solve the system of linear equationsusing the substitution method. (See Example 4.)19. 2x y 6z 120. x 6y 2z 83x 2y 5z 167x 3y 4z 11 x 5y 3z 23x 2y 4z 1821. x y z 45x 5y 5z 12x 4y z 922. x 2y 1 x 3y 2z 4 x y 4z 1031. WRITING Explain when it might be more convenientto use the elimination method than the substitutionmethod to solve a linear system. Give an example tosupport your claim.32. REPEATED REASONING Using what you know aboutsolving linear systems in two and three variables, plana strategy for how you would solve a system that hasfour linear equations in four variables.MATHEMATICAL CONNECTIONS In Exercises 33 and 34,write and use a linear system to answer the question.23. 2x 3y z 10y 2z 13z 525. x y z 43x 2y 4z 17 x 5y z 827. 4x y 5z 58x 2y 10z 10x y 2z 224. x 4x y 64x 3y 2z 2626. 2x y z 154x 5y 2z 10 x 4y 3z 2028. x 2y z 32x 4y 2z 6 x 2y z 633. The triangle has a perimeter of 65 feet. What are thelengths of sidesℓ, m, and n?n 13 m m 15m34. What are the measures of angles A, B, and C?A29. PROBLEM SOLVING The number of left-handedA people in the world is one-tenth the number of righthanded people. The percent of right-handed peopleis nine times the percent of left-handed people andambidextrous people combined. What percent ofpeople are ambidextrous?(5A C) (A B) BC35. OPEN-ENDED Consider the system of linearequations below. Choose nonzero values for a, b,and c so the system satisfies the given condition.Explain your reasoning.30. MODELING WITH MATHEMATICS Use a system oflinear equations to model the data in the followingnewspaper article. Solve the system to find how manyathletes finished in each place.Lawrence High prevailed in Saturday’s trackmeet with the help of 20 individual-eventplacers earning a combined 68 points. Afirst-place finish earns 5 points, a secondplace finish earns 3 points, and a third-placefinish earns 1 point. Lawrence had a strongsecond-place showing, with as many secondplace finishers as first- and third-placefinishers combined.x y z 2ax by cz 10x 2y z 4a. The system has no solution.b. The system has exactly one solution.c. The system has infinitely many solutions.36. MAKING AN ARGUMENT A linear system in threevariables has no solution. Your friend concludes that itis not possible for two of the three equations to haveany points in common. Is your friend correct? Explainyour reasoning.Section 1.4hsnb alg2 pe 0104.indd 35Solving Linear Systems352/5/15 9:57 AM
37. PROBLEM SOLVING A contractor is hired to build an40. HOW DO YOU SEE IT? Determine whether theapartment complex. Each 840-square-foot unit has abedroom, kitchen, and bathroom. The bedroom willbe the same size as the kitchen. The owner orders980 square feet of tile to completely cover the floorsof two kitchens and two bathrooms. Determinehow many square feet of carpet is needed for eachbedroom.BATHROOMsystem of equations that represents the circles hasno solution, one solution, or infinitely many solutions.Explain your reasoning.a.b.yyxKITCHENx41. CRITICAL THINKING Find the values of a, b, and c sothat the linear system shown has ( 1, 2, 3) as itsonly solution. Explain your reasoning.BEDROOMTotal Area: 840 ft2x 2y 3z a x y z b2x 3y 2z c38. THOUGHT PROVOKING Does the system of linearequations have more than one solution? Justifyyour answer.42. ANALYZING RELATIONSHIPS Determine whicharrangement(s) of the integers 5, 2, and 3 producea solution of the linear system that consist of onlyintegers. Justify your answer.4x y z 02x —12 y 3z 0 x —14y z 0x 3y 6z 21x y z 302x 5y 2z 639. PROBLEM SOLVING A florist must make 5 identicalbridesmaid bouquets for a wedding. The budget is 160, and each bouquet must have 12 flowers. Rosescost 2.50 each, lilies cost 4 each, and irises cost 2 each. The florist wants twice as many roses as theother two types of flowers combined.43. ABSTRACT REASONING Write a linear system torepresent the first three pictures below. Use the systemto determine how many tangerines are required tobalance the apple in the fourth picture. Note: Thefirst picture shows that one tangerine and one applebalance one grapefruit.a. Write a system of equations to represent thissituation, assuming the florist plans to use themaximum budget.0 10 20 30 40 50 60 70 80 90 100 1100 120 130 140 150 160 170 180 190 2000 10 20 30 40 50 60 70 80 90 100 1100 120 130 140 150 160 170 180 190 2000 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200b. Solve the system to find how many of each type offlower should be in each bouquet.c. Suppose there is no limitation on the total cost ofthe bouquets. Does the problem still have exactlyone solution? If so, find the solution. If not, givethree possible solutions.0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200Maintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsSimplify. (Skills Review Handbook)44.(x 2)245.(3m 1)246.(2z 5)2Write a function g described by the given transformation of f(x) x 5.3647.(4 y)2(Section 1.2)48. translation 2 units to the left49.reflection in the x-axis50. translation 4 units up51.vertical stretch by a factor of 3Chapter 1hsnb alg2 pe 0104.indd 36Linear Functions2/5/15 9:57 AM
Section 1.4 Solving Linear Systems 31 Solving Systems of Equations Algebraically The algebraic methods you used to solve systems of linear equations in two variables can be extended to solve a system of linear equations in three variables. Solving a Three-Variable System (On
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If you have a linear system Ax b and B is an inverse matrix for A then the linear system has the unique solution x Bb: Solving Linear Systems Math 240 Solving Linear Systems Gauss-Jordan elimination . Solve the linear system x 1 3 2 1; 2x 1 5x 2 3: The coe cient matrix is A 1 3 2 5 , so
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