4.4 Solving Systems of Equations by Matrices

Transcription

1 Setion 4.4 Solving Systems of Equations by Matries 1. A first number is 8 less than a seond number. Twie the first number is 11 more than the seond number. Find the numbers.. The sum of the measures of the angles of a quadrilateral is 6. The two smallest angles of the quadrilateral have the same measure. The third angle measures more than the measure of one of the smallest angles and the fourth angle measures more than the measure of one of the smallest angles. Find the measure of eah angle. 4.4 Solving Systems of Equations by Matries S 1 Use Matries to Solve a System of Two Equations. Use Matries to Solve a System of Three Equations. Helpful Hint Before writing the orresponding matri assoiated with a system of equations, make sure that the equations are written in standard form. By now, you may have notied that the solution of a system of equations depends on the oeffiients of the equations in the system and not on the variables. In this setion, we introdue solving a system of equations by a matri. 1 Using Matries to Solve a System of Two Equations A matri (plural: matries) is a retangular array of numbers. The following are eamples of matries. J R C -1 4 S J a b d e f R -6 1 The numbers aligned horizontally in a matri are in the same row. The numbers aligned vertially are in the same olumn. row 1 S row S J R This matri has rows and olumns. It is alled a * (read ;two by three<) matri. olumn 1 u olumn æ olumn To see the relationship between systems of equations and matries, study the eample below. System of Equations (in standard form) - y = 6 b + y = Equation 1 Equation Corresponding Matri J R Row 1 Row Notie that the rows of the matri orrespond to the equations in the system. The oeffiients of eah variable are plaed to the left of a vertial dashed line. The onstants are plaed to the right. Eah of the numbers in the matri is alled an. The method of solving systems by matries is to write this matri as an equivalent matri from whih we easily identify the solution. Two matries are equivalent if they represent systems that have the same solution set. The following row operations an be performed on matries, and the result is an equivalent matri. Elementary Row Operations 1. Any two rows in a matri may be interhanged.. The s of any row may be multiplied (or divided) by the same nonzero number.. The s of any row may be multiplied (or divided) by a nonzero number and added to their orresponding s in any other row. Helpful Hint Notie that these row operations are the same operations that we an perform on equations in a system.

2 6 CHAPTER 4 Systems of Equations To solve a system of two equations in and y by matries, write the orresponding matri assoiated with the system. Then use ary row operations to write equivalent matries until you have a matri of the form J 1 a b 1 R, where a, b, and are onstants. Why? If a matri assoiated with a system of equations is in this form, we an easily solve for and y. For eample, Matri System of Equations J 1 1 PRACTICE y = - R orresponds to b + 1y = First equation Let y =. The ordered pair solution is 1-1,. Chek to see that this ordered pair satisfies both equations. 1 Use matries to solve the system. + 4y = - e - y = 7 + y = - or b y = In the seond equation, we have y =. Substituting this in the first equation, we have + 1 = - or = -1. The solution of the system is the ordered pair 1-1,. EXAMPLE 1 Use matries to solve the system. b + y = - y = -4 Solution The orresponding matri is J 1-1 R. We use ary row -4 operations to write an equivalent matri that looks like J 1 a b 1 R. For the matri given, the in the first row, first olumn is already 1, as desired. Net we write an equivalent matri with a below the 1. To do this, we multiply row 1 by - and add to row. We will hange only row. 1 J row 1 row row 1 row 1 R simplifies to J row 1 row -14 R Now we hange the -7 to a 1 by use of an ary row operation. We divide row by -7, then 1 C S simplifies to J 1 1 R This last matri orresponds to the system e + y = y = To find, we let y = in the first equation, + y =. + y = + 1 = = -1

3 Setion 4.4 Solving Systems of Equations by Matries 7 EXAMPLE Use matries to solve the system. e - y = 4 - y = Solution The orresponding matri is J position, we divide the s of row 1 by. R. To get 1 in the row 1, olumn simplifies to To get under the 1, we multiply the s of row 1 by -4 and add the new s to the s of row. 1-1 D a b a T simplifies to b + -1 The orresponding system is - 1 y =. The equation = -1 is false for all y or = -1 values; hene, the system is inonsistent and has no solution. PRACTICE Use matries to solve the system. - y = e - + 6y = 4 CONCEPT CHECK Consider the system - y = 8 e + y = - What is wrong with its orresponding matri shown below? J - 8 ` 8 - R Using Matries to Solve a System of Three Equations To solve a system of three equations in three variables using matries, we will write the orresponding matri in the form 1 a b C 1 1 d e S f Answer to Conept Chek: matri should be J R EXAMPLE (Continued on net page) Use matries to solve the system. + y + z = - - y + z = + y - z = -8

4 8 CHAPTER 4 Systems of Equations 1 1 Solution The orresponding matri is C - -1 S. Our goal is to write an equivalent matri with 1 s along the diagonal (see the numbers in red) and s below the 1 s. The in row 1, olumn 1 is already 1. Net we get s for eah in the rest of olumn 1. To do this, first we multiply the s of row 1 by and add the new s to row. Also, we multiply the s of row 1 by -1 and add the new s to the s of row. We do not hange row 1. Then C S simplifies to C S -1 We ontinue down the diagonal and use ary row operations to get 1 where the is now. To do this, we interhange rows and. 1 1 C S -1 is equivalent to C S 9 Net we want the new row, olumn to be. We multiply the s of row by - and add the result to the s of row. 1 1 C C S 9-1 S simplifies to Finally, we divide the s of row by 1 so that the final diagonal is D T simplifies to C S This matri orresponds to the system + y + z = y - z = -1 z = We identify the z-oordinate of the solution as. Net, we replae z with in the seond equation and solve for y. y - z = - 1 Seond equation y - 1 = -1 Let z =. y = -1 To find, we let z = and y = -1 in the first equation. + y + z = First equation = Let z = and y = -1. = 1 The ordered triple solution is 11, -1,. Chek to see that it satisfies all three equations in the original system.

5 Setion 4.4 Solving Systems of Equations by Matries 9 PRACTICE Use matries to solve the system. + y - z = + y + z = - - y + 4z = 7 Voabulary, Readiness & Video Chek Use the hoies below to fill in eah blank. olumn row matri 1. A(n) is a retangular array of numbers.. Eah of the numbers in a matri is alled a(n).. The numbers aligned horizontally in a matri are in the same. 4. The numbers aligned vertially in a matri are in the same. Answer true or false for eah statement about operations within a matri forming an equivalent matri.. Any two olumns may be interhanged. 6. Any two rows may be interhanged. 7. The s in a row may be added to their orresponding s in another row. 8. The s of a olumn may be multiplied by any nonzero number. Martin-Gay Interative Videos Wath the setion leture video and answer the following questions From the leture before Eample 1, what ary row operations an be performed on matries? Whih operation is not performed during Eample 1? 1. In Eample, why do you think the suggestion is made to write neatly when using matries to solve systems? See Video Eerise Set Solve eah system of linear equations using matries. See Eample 1. Solve eah system of linear equations using matries. See Eample. 1. e + y = 1 - y = 4. e + y = + y =. e - y = 4-4y = 4 - y = 9 7. e - y = 6. e - y = 8 + y = e 4 - y = + y = Solve eah system of linear equations using matries. See Eample. - + y = 6 6. e - 9y = y = 6 8. e y = -1 + y = 9. y = 1 + y - 4z = 1 y - z = y + z = -4 - y + z = 1 MIXED PRACTICE = 1. + y = 4 + y - z = y + z = - - 4y = y + z = - Solve eah system of linear equations using matries. See Eamples 1 through. 1. e - 4 = y = e + y = 1 + y = 7

6 4 CHAPTER 4 Systems of Equations + y + z = 1. - z = y + z = + y + z = y - z = y + z = 1 1 a. C S 1 1 b. C S - y = e - + y = y = e 1-1y = y + z = 1. y + z = y + z = y + z = - + y - z = y + z = -1 REVIEW AND PREVIEW 18. e 4 - y = 9 + y = -7 - y = 1. e y = y - 7z = y + 4z = 1 + 6y - z = 4. + y + z = 9 - y + z = y - z = - Determine whether eah graph is the graph of a funtion. See Setion... y6. y 7. Evaluate. See Setion CONCEPT EXTENSIONS y8. Solve. See the Conept Chek in the setion. + z = 7. For the system y + z = -6, whih is the orret - y = orresponding matri? y 1 1. C S 6. For the system e - 6 =, whih is the orret orresponding - y = 1 matri? a. J R b. J1 1-6 R. J R 7. The amount of eletriity y generated by geothermal soures (in billions of kilowatts) from to 9 an be modeled by the linear equation y -.11 = 14., where represents the number of years after. Similarly, the amount of eletriity y generated by wind power (in billions of kilowatts) during the same time period an be modeled by the linear equation.1 - y =.6. (Soure: Based on data from Energy Information Administration, U.S. Department of Energy) a. The data used to form these two models were inomplete. It is impossible to tell from the data the year in whih the eletriity generated by geothermal soures was the same as the eletriity generated by wind power. Use matri methods to estimate the year in whih this ourred. b. The earliest data for wind power was in 1989, where.1 billion kilowatts of eletriity was generated. Can this data be determined from the given equation? Why do you think that is?. Aording to these models, will the perent of eletriity generated by geothermal ever go to zero? Why? d. Can you think of an eplanation why the amount of eletriity generated by wind power is inreasing so muh faster than the amount of eletriity generated by geothermal power? 8. The most popular amusement park in the world (aording to attendane) is Walt Disney World s Magi Kingdom, whose yearly attendane in millions an be approimated by the equation y = where is the number of years after. In seond plae is Walt Disney World s Disneyland, whose yearly attendane in millions an be approimated by the equation y = where is the number of years after. Find the year when attendane in Disneyland is equal to the attendane in Magi Kingdom. (Soure: Themed Entertainment Assoiation, Eonomis Researh Assoiates) 9. For the system e - y = 8, eplain what is wrong with + y = - writing the orresponding matri as J 8 - R. + y = 4. For the system e, eplain what is wrong with -y = writing the orresponding matri as J -1 R.