4.5 Solving Systems Using Inverse Matrices - …

1 Page 1 of 2230 Chapter 4 Matrices and DeterminantsSolving Systems Using Inverse MatricesSOLVINGSYSTEMSUSINGMATRICESIn Lesson you learned how to solve a system of linear equations Using Cramer srule. Here you will learn to solve a system Using Inverse the activity you learned that a linear system can be written as a matrix equation AX= B. The matrix Ais the coefficient matrix of the system , X is the andBis the Writing a Matrix EquationWrite the system of linear equations as a matrix equation. 3x+ 4y= 5 Equation 12x y = 10 Equation 2 SOLUTIONAX B = ..Once you have written a linear system as AX= B, you can solve for Xby multiplyingeach side of the matrix by A 1on the original matrix 1AX= A 1 BMultiply each side by A A 1BA 1A=IX= A 1 BIX=X5 10xy4 1 32 EXAMPLE 1matrix of of variables,GOAL1 Solve Systems oflinear equations usinginverse Systems oflinear equations to solvereal-lifeproblems, such asdetermining how muchmoney to invest in Example 4.

2 To solve real-lifeproblems, such as planning a stained glass project inEx. you should learn itGOAL2 GOAL1 What you should Matrix EquationsWrite the left side of the matrix equation as a single matrix. Then = Matrix equationequate corresponding entries of the Matrices . What do you obtain?Use what you learned in Step 1to 2x y= 4 Equation 1write the following linear system 4x+ 9y= 1 Equation 2as a matrix 42511 DevelopingConceptsACTIVITYPage 1 of Systems Using Inverse Matrices231 SOLUTION OF ALINEARSYSTEMLet AX= Brepresent a system of linearequations. If the determinant of Ais nonzero, then the linear system has exactly one solution, which is X= A a Linear SystemUse Matrices to solve the linear system in Example 1. 3x+ 4y = 5 Equation 12x y = 10 Equation 2 SOLUTIONB egin by writing the linear system in matrix form, as in Example 1. Then find theinverse of matrix 1= 3 18 = Finally, multiply the matrix of constants by A A 1B= = = The solution of the system is ( 7, 4).

3 Check this solution in the a Graphing CalculatorUse a matrix equation and a graphing calculator to solve the linear + 3y+z= 1 Equation 13x+ 3y+ z= 1 Equation 22x+ 4y+ z= 2 Equation 3 SOLUTIONThe matrix equation that represents the system is = . Using a graphing calculator, you can solve the system as matrix matrix Bby A 1. The solution is (2, 1, 2). Check this solution in the original equations.[A]-1[B][[2 ][-1][-2]]MATRIX [B] 3X1[-1][1][-2 ]MATRIX [A] 3X3[231][331][241] 11 2xyz111334232 EXAMPLE 3xy 7 45 10 54 35 51 25 54 35 51 25 4 3 1 2 EXAMPLE 2 STUDENTHELPS tudy TipRemember that you canuse the method shown inExamples 2 and 3 providedAhas an Inverse . If Adoes not have an Inverse ,then the system has eitherno solution or infinitelymany solutions, and youshould use a Back For help with Systems of linear equations, see p. 1 of 2 USINGLINEARSYSTEMS INREALLIFEW riting and Using a Linear SystemINVESTINGYou have $10,000 to invest.

4 You want to invest the money in a stockmutual fund, a bond mutual fund, and a money market fund. The expectedannual returns for these funds are given in the want your investment to obtain an overall annual return of 8%. A financial planner recommends that you invest the same amount in stocks as in bonds and the money market combined. How much should you invest in each fund?SOLUTION++ = + + + Stock amount = Money market amount = Bond amount = Total invested = 10,000+ + = 10,000 Equation + + (10,000)Equation 2= + Equation 3 First rewrite the equations above in standard form and then in matrix +b+m = 10, + + 800 = s b m= 0 Enter the coefficient matrix Aand the matrix of constants Binto a graphingcalculator. Then find the solution X= A 1B. You should invest $5000 in the stock mutual fund, $2500 in the bond mutual fund,and $2500 in the money market fund.[A]-1[B] [[5000] [2500] [2500]]MATRIX [B] 3X1[10000][800][0]MATRIX [A] 3X3[111 ][.]

5 ][1-1-1]10, 4 GOAL2232 Chapter 4 Matrices and DeterminantsINVESTING Each year studentsacross the country in grades4 through 12 invest ahypothetical $100,000 instocks to compete in theStock Market Game. Studentscan enter their transactionsusing the ONAPPLICATIONSALGEBRAICMODELLABELSVERBAL MODELI nvestmentExpected returnStock mutual fund10%Bond mutual fund7%Money market (MM) fund 5%Page 1 of Systems Using Inverse are a matrix of variables and a matrix of constants, and how are they usedto solve a system of linear equations? |A| 0, what is the solution of AX= Bin terms of Aand B? why the solution of AX = Bis not X= BA the linear system as a matrix + y= + 3y= + y+ z= 102x y= 64x 2y= 75x y= 13x+ 4y+ z= 8 Use an Inverse matrix to solve the linear +y= 28. x 2y= +3y= 67x+8y= 212x+8y= 16x 2y= back at Example 4 on page 232. Suppose you have $60,000 to invest and you want an overall annual return of 9%.

6 Use the expected annual returns shown to determine how much you should invest in each fund. Assume you are investing as much in stocks as in bonds and the money market the linear system as a matrix + y= + 2y= 3y= 93x 4y= 84x y = 5 4x + 2y= 5y= + 8y= 5y= 4 3x+ 7y= 154x 5y = 11x 3y= 4y+ 5z= y+ 4z= + + y 7z = 23 2x+ 4y z = + 4x + 5y + 2z= 38x y+ 3z = + +z= 10z = +y z = 0 x y+2z=66y 12z=142x z= 12x+7y z = 4 9x+5z=0y+ z= 2 SOLVINGSYSTEMSUse an Inverse matrix to solve the linear + y = + y = + 7y= 535x+ 2y= 1111x+ 12y= 8x + 3y= + 5y= 7y= 5428. 5x 7y= 94x + 3y= 42x 4y= 302x+ 3y= + 2y = + 4y = 5y= 43 2x 3y= 142x+ 5y= 31 2x+ 2y= 22 PRACTICEANDAPPLICATIONSGUIDEDPRACTICEV ocabulary Check Concept Check Skill Check InvestmentExpected returnStock mutual fund12%Bond mutual fund8%Money market fund5%Extra Practiceto help you masterskills is on p. HELPE xample 1:Exs.

7 11 22 Example 2:Exs. 23 31 Example 3:Exs. 32 39 Example 4:Exs. 40 44 Page 1 of 2234 Chapter 4 Matrices and DeterminantsSOLVINGSYSTEMSUse the given Inverse of the coefficient matrix to solvethe linear z = y 3z= 95x+2y+ 3z= 45x+ 2y+z= 307x+3y+4z= 5 3x y= 4A 1= A 1= SOLVINGSYSTEMSUse an Inverse matrix and a graphing calculator tosolve the linear + 2y= 1335. x+ y 3z = + 5y 5z = 213x + 2y+z = 133x 2y + 8z= 14 4x + 8y 5z = 12x+ y + 3z = 92x 2y+ 5z= 72x 5y+ 6z= + z = + 3y+ z= + y 3z= 175x y+z = 56x + y= 92x+ z = 12 x+ 2y + 2z= 03x+ 5y + 3z = 21 7x 2y +z = are planning a birthday party for your youngerbrother at a skating rink. The cost of admission is $ per adult and $ perchild, and there is a limit of 20 people. You have $50 to spend. Use an inversematrix to determine how many adults and how many children you can use various amalgams for silver fillings. Thematrix shows the percents (expressed as decimals) of powdered alloys used inpreparing three different amalgams.

8 Suppose a dentist has 5483 grams of silver,2009 grams of tin, and 129 grams of copper. How much of each amalgam can be made?PERCENTALLOY BYWEIGHTA malgamABC are making mosaic tiles from three types of stainedglass. You need 6 square feet of glass for the project and you want there to be asmuch iridescent glass as red and blue glass combined. The cost of a sheet ofglass having an area of square foot is $ for iridescent, $ for red, and$ for blue. How many sheets of each type should you purchase if you plan tospend $45 on the project? walkway lighting package includes a transformer,a certain length of wire, and a certain number of lights on the wire. The price ofeach lighting package depends on the length of wire and the number of lights onthe wire. A package that contains a transformer, 25 feet of wire, and 5 lights costs $20. A package that contains a transformer, 50 feet of wire, and 15 lights costs $35.

9 A package that contains a transformer, 100 feet of wire, and 20 lights costs $ and solve a system of equations to find the cost of a transformer, the costper foot of wire, and the cost of a light. Assume the cost of each item is the samein each lighting 1673 941 318 5 10 11714 111 DENTISTD entists diagnose,prevent, and treat problemsof the teeth and amalgams have beenused for more than 150 yearsto restore the teeth of over100 million ONCAREERSHOMEWORK HELPV isit our Web help with Exs. 32 and 1 of Systems Using Inverse are an accountant for a constructionbusiness and are planning next year s budget. You have $200,000 to spend onsalaries, equipment maintenance, and other general expenses. Based on previousfinancial records of the business, you expect to spend five times as much onsalaries as on equipment maintenance, and you expect general expenses to be10% of the amount spent on the other two categories combined.

10 Write and solvea system of equations to find the amount you should budget for each company sells different sizes of gift baskets witha varying assortment of meat and cheese. A basic basket with 2 cheeses and 3 meats costs $15, a big basket with 3 cheeses and 5 meats costs $24, and asuper basket with 7 cheeses and 10 meats costs $ and solve a system of equations Using the information about the basicand big and solve a system of equations Using the information about the big andsuper the results from parts (a) and (b) and make a conjectureabout why there is a OFFOUREQUATIONSS olve the linear system Using thegiven Inverse of the coefficient + 6x+ 3y 3z = 22w+ 7x+ y+ 2z= 5w+ 5x + 3y 3z= 3 6x 2y + 3z= 6 EVALUATINGFUNCTIONSE valuate (x) or g(x) for the given value of x. (Review ) (x) = g(x) = 47. (8)48. (11)49. ( 2)50. (0) (3) (0) ( 1) ( 3)GRAPHINGFUNCTIONSG raph the function and label the vertex. (Review for ) |x 5| |x| + |x 8| |x 5| + |x + 3| + |x+ 6| 2 FINDINGINVERSESFind the Inverse of the matrix.