Chapter 11 Systems of Equations and Inequalities. ( 2,4), is a solution of the system of equations. 10. ( 1,2

Transcription

1 Chapter Sstems of Equations and Inequalities Section The solution set is { }.. a. + -intercept: + ( ) -intercept: ( ) + b inconsistent A parallel line would have slope. consistent; independent. (, ) 6. consistent; dependent Substituting the values of the variables: () ( ) + () + ( ) Each equation is satisfied, so,, or (, ), is a solution of the sstem of equations. + 7 Substituting the values of the variables: ( ) + () ( ) 7() 8 Each equation is satisfied, so,, or (,), is a solution of the sstem of equations. Substituting the values of the variables: () 6 () Each equation is satisfied, so,, or (, ), is a solution of the sstem of equations. + 9 Substituting the values of the variables, we obtain: + ( ) + 9 ( ) 8 Each equation is satisfied, so,, or (, ), is a solution of the sstem of equations. + Substituting the values of the variables, we obtain: 7 Copright Pearson Education, Inc. Publishing as Prentice Hall

2 Section.: Sstems of Linear Equations: Substitution and Elimination.... () + + Each equation is satisfied, so,, or (,), is a solution of the sstem of equations. + Substituting the values of the variables: ( ) ( ) + ( ) + ( ) 6 Each equation is satisfied, so,, or (, ), is a solution of the sstem of equations. + + z z z 8 Substituting the values of the variables: () + ( ) + () + ( ) + ( ) () 6 8 Each equation is satisfied, so,, z, or (,, ), is a solution of the sstem of equations. z 7 8+ z + z 6 Substituting the values of the variables: ( ) ( ) + ( ) 6 ( ) + ( ) Each equation is satisfied, so,, z, or (,,), is a solution of the sstem of equations. + + z + z z 8 Substituting the values of the variables: ( ) + ( ) + ( ) 6 6+ ( ) ( ) ( ) ( ) Each equation is satisfied, so,, z, or (,, ) is a solution of the sstem of equations. z 6 z z Substituting the values of the variables: ( ) ( ) 6 6 ( ) ( ) 7 ( ) 6( ) + ( ) Each equation is satisfied, so,, z, or (,, ), is a solution of the sstem of equations. + 8 Solve the first equation for, substitute into the second equation and solve: 8 (8 ) 8+ 6 Since 6, 8 6. The solution of the sstem is 6, or using ordered pairs (6, ) Solve the first equation for, substitute into the second equation and solve: 7 + ( 7 ) + 7 Since, 7( ). The solution of the sstem is, or using an ordered pair, (, ). 7 Copright Pearson Education, Inc. Publishing as Prentice Hall

3 Chapter : Sstems of Equations and Inequalities 9. + Multipl each side of the first equation b and add the equations to eliminate : Substitute and solve for : () 6 6 The solution of the sstem is, 6 or using ordered an pair (, 6) Solve the second equation for and substitute into the first equation: + + () + The solution of the sstem is, or using ordered pairs, Add the equations: + 8 Substitute and solve for : + 6 The solution of the sstem is, or using ordered pairs (,). + Solve the first equation for and substitute into the second equation: Multipl each side of the first equation b and each side of the second equation b, then add to eliminate : Substitute and solve for : / ( ) The solution of the sstem is, or 6 using ordered pairs, The solution of the sstem is 8, or using ordered pairs (8, ) 76 Copright Pearson Education, Inc. Publishing as Prentice Hall

4 Section.: Sstems of Linear Equations: Substitution and Elimination Multipl each side of the first equation b and each side of the second equation b, then add to eliminate : + Substitute and solve for : ( /) The solution of the sstem is, or using ordered pairs,. + + Solve the first equation for, substitute into the second equation and solve: + + ( ) + This equation is false, so the sstem is inconsistent. + Solve the first equation for, substitute into the second equation and solve: + + ( + ) This equation is false, so the sstem is inconsistent Solve the first equation for, substitute into the second equation and solve: + + ( ) + 8 Since, The solution of the sstem is, or using ordered pairs, Solve the second equation for, substitute into the first equation and solve: Since, (). The solution of the sstem is, or using ordered pairs, Solve the first equation for, substitute into the second equation and solve: Copright Pearson Education, Inc. Publishing as Prentice Hall

5 Chapter : Sstems of Equations and Inequalities.. ( ) These equations are dependent. The solution of the sstem is either, where is an real number or, where is an real number. Using ordered pairs, we write the solution as (, ), is an real number or as { } (, ), is an real number. 7 9 Solve the first equation for, substitute into the second equation and solve: 7 9 9( 7) 9 9+ These equations are dependent. The solution of the sstem is either 7, where is an real + 7 number is, where is an real number. Using ordered pairs, we write the solution as (, ) 7, is an real number or as { } 7 (, ) +, is an real number. + Multipl each side of the first equation b, and add the equations to eliminate : Substitute and solve for : () The solution of the sstem is, or using ordered pairs (, ). 78 Copright Pearson Education, Inc. Publishing as Prentice Hall.. + Multipl each side of the first equation b, and add the equations to eliminate : + Substitute and solve for : / ( ) + + The solution of the sstem is, or using ordered pairs,. + 6 Solve the second equation for, substitute into the first equation and solve: Since, +. The solution of the sstem is, or using ordered pairs,.

6 Section.: Sstems of Linear Equations: Substitution and Elimination Solve the second equation for, substitute into the first equation and solve: ( + 8) Since, ( ) The solution of the sstem is, or using ordered pairs (, ). + Multipl each side of the first equation b 6 and each side of the second equation b, then add to eliminate : 8 8 Substitute and solve for : + () + The solution of the sstem is, or using ordered pairs (, ) Multipl each side of the first equation b and each side of the second equation b, then add to eliminate : Substitute and solve for : + (6) + 9 The solution of the sstem is, 6 or using ordered pairs (, 6). + Add the equations to eliminate : + 8 Substitute and solve for : / ( ) The solution of the sstem is, or using ordered pairs,. 79 Copright Pearson Education, Inc. Publishing as Prentice Hall

7 Chapter : Sstems of Equations and Inequalities Multipl each side of the second equation b, and add the equations to eliminate : + Substitute and solve for : The solution of the sstem is using ordered pairs,. + 8 Rewrite letting u, v :, or u+ v 8 u v Solve the first equation for u, substitute into the second equation and solve: u 8 v u v (8 v) v v v 8v v Since v, u 8. Thus, u, v. The solution of the sstem is, or using ordered pairs, Rewrite letting u, v : u v 6u+ v Multipl each side of the second equation b, and add the equations to eliminate v: u v u+ v 6 u u 6 Substitute and solve for v: v v v v Thus,, u v. The solution of the sstem is, or using ordered pairs (, ). 6 z 6 + z Multipl each side of the first equation b and add to the second equation to eliminate : + z 6 z Multipl each side of the result b and add to the original third equation to eliminate : + z + z z z Substituting and solving for the other variables: 8 Copright Pearson Education, Inc. Publishing as Prentice Hall

8 Section.: Sstems of Linear Equations: Substitution and Elimination + () The solution is 8,, z or using ordered triples (8,, ). z 8 + 7z z z Substituting and solving for the other variables: z z Multipl each side of the first equation b and add to the second equation to eliminate : z + z 8 Multipl each side of the result b and add to the original third equation to eliminate z: + z z Substituting and solving for the other variables: ( ) + ( ) z 6+ 9 z z z The solution is,, z or using ordered triples (,,). + z z + z Multipl each side of the first equation b and add to the second equation to eliminate ; and multipl each side of the first equation b and add to the third equation to eliminate : + 6z + + z z 6+ 9z + z + 7z. ( ) + () The solution is,, z or using ordered triples (,,). + z + + z 7 z 7 Multipl each side of the first equation b and add to the second equation to eliminate ; and multipl each side of the first equation b and add to the third equation to eliminate : + 6z + + z z 7 8+ z z 7 z 7 Multipl each side of the first result b and multipl each side of the second result b 6 to eliminate : z z z z Substituting and solving for the other variables: Multipl each side of the first result b and add to the second result to eliminate : 8 Copright Pearson Education, Inc. Publishing as Prentice Hall

9 Chapter : Sstems of Equations and Inequalities The solution is,, z or 6 7 using ordered triples,,. z + + z + Add the first and second equations to eliminate z: z + + z + Multipl each side of the result b and add to the original third equation to eliminate : + This equation is false, so the sstem is inconsistent. z + + z z Add the first and second equations to eliminate z; then add the second and third equations to eliminate z: z + + z + + z z 6 Multipl each side of the first result b and add to the second result to eliminate : + 6 This equation is false, so the sstem is inconsistent z + z 7z Add the first and second equations to eliminate ; multipl the first equation b and add to the third equation to eliminate : z + z z + + z 7z z Multipl each side of the first result b and add to the second result to eliminate : + z z The sstem is dependent. If z is an real number, then z. Solving for in terms of z in the first equation: (z) z z+ z z+ z The solution is {( z,, ) z, z, z is an real number}. z + + z + + z Multipl the first equation b and add to the second equation to eliminate z; multipl the first equation b and add to the third equation to eliminate z: 6 z + + z 7 69 z + + z 7 Multipl each side of the first result b and add to the second result to eliminate : 8 Copright Pearson Education, Inc. Publishing as Prentice Hall

10 Section.: Sstems of Linear Equations: Substitution and Elimination The sstem is dependent. If is an real number, then Solving for z in terms of in the first equation: z The solution is ( z,, ) +, 7 7 z +, is an real number z 6 + z + 7z Multipl the first equation b and add to the second equation to eliminate ; add the first and third equations to eliminate : + 6z + z z + z 6 + 7z z 7 Multipl each side of the first result b and add to the second result to eliminate : + z z 7 9 This result is false, so the sstem is inconsistent... + z z + z Multipl the first equation b and add to the second equation to eliminate z; multipl the first equation b and add to the third equation to eliminate z: + z z 7 6+ z + z + Add the first result to the second result to eliminate : This result is false, so the sstem is inconsistent. + z 6 + z + z Add the first and second equations to eliminate z; multipl the second equation b and add to the third equation to eliminate z: + z 6 + z 6 + z + z 7 Multipl each side of the first result b and add to the second result to eliminate : + 7 Substituting and solving for the other variables: () () () + z 6+ z z The solution is,, z or using ordered triplets (,, ). 8 Copright Pearson Education, Inc. Publishing as Prentice Hall

11 Chapter : Sstems of Equations and Inequalities.. + z + z + z Multipl the first equation b and add to the second equation to eliminate ; add the first and third equations to eliminate : + z + z + z z + z + z 6 z 8 Substitute and solve: 6 ( ) z + z () + ( ) The solution is,, z or using ordered triplets (,, ). + z + z 7 + z Add the first and second equations to eliminate z; multipl the second equation b and add to the third equation to eliminate z: + z + z z + z 7 Multipl each side of the first result b and add to the second result to eliminate : + 7 Substituting and solving for the other variables: ( ) z + z z z The solution is,, z or using ordered triplets,,. Copright Pearson Education, Inc. Publishing as Prentice Hall. + z 8 + z + + 6z Add the first and second equations to eliminate z; multipl the first equation b and add to the third equation to eliminate z: + z 8 + z z z + 9 Multipl each side of the second result b / and add to the first result to eliminate : + 9 Substituting and solving for the other variables: z 6z z 9 8 The solution is,, z or using 9 8 ordered triplets,, 9.

12 Section.: Sstems of Linear Equations: Substitution and Elimination. Let l be the length of the rectangle and w be the width of the rectangle. Then: l w and l+ w 9 Solve b substitution: ( w) + w 9 w+ w 9 6w 9 w feet l () feet The floor is feet b feet. 6. Let l be the length of the rectangle and w be the width of the rectangle. Then: l w+ and l+ w Solve b substitution: ( w+ ) + w w+ + w w 9 w 7 meters l meters The dimensions of the field are 77 meters b 7 meters. 7. Let the number of commercial launches and the number of noncommercial launches. Then: + and + Solve b substitution: + (+ ) (8) In there were 8 commercial launches and 7 noncommercial launches. 8. Let the number of adult tickets sold and the number of senior tickets sold. Then: Solve the first equation for : Solve b substitution: 9+ 7( ) There were adult tickets sold and senior citizen tickets sold. 9. Let the number of pounds of cashews. Let is the number of pounds in the miture. The value of the cashews is. The value of the peanuts is.(). The value of the miture is. Then + represents the amount of miture. + represents the value of the miture. Solve b substitution: + ( + ). So,. pounds of cashews should be used in the miture. 6. Let the amount invested in AA bonds. Let the amount invested in the Bank Certificate. a. Then +, represents the total investment..+., represents the earnings on the investment. Solve b substitution:.(, ) +.,,.+.,. 6,, 6, 9, Thus, $9, should be invested in AA Bonds and $6, in a Bank Certificate. b. Then +, represents the total investment..+., represents the earnings on the investment. Solve b substitution:.(, ) +.,,.+.,.,,,, Thus, $, should be invested in AA Bonds and $, in a Bank Certificate. 8 Copright Pearson Education, Inc. Publishing as Prentice Hall

13 Chapter : Sstems of Equations and Inequalities 6. Let the plane s average airspeed and the average wind speed. Rate Time Distance With Wind + 6 Against 6 ( + )() 6 ( )() 6 Multipl each side of the first equation b, multipl each side of the second equation b, and add the result to eliminate The average airspeed of the plane is 7 mph, and the average wind speed is mph. 6. Let the average wind speed and the distance. Rate Time Distance With Wind + Against ( + )() ( )() Solve b substitution: ( + )() ( )() + Thus, the average wind speed is mph. 6. Let the number of $-design. Let the number of $-design. Then + the total number of sets of dishes. + the cost of the dishes. Setting up the equations and solving b substitution: Solve the first equation for, the solve b substitution: + ( ) Thus, 8 sets of the $ dishes and sets of the $ dishes should be ordered. 6. Let the cost of a hot dog. Let the cost of a soft drink. Setting up the equations and solving b substitution: (7 ) (.7). A single hot dog costs $.7 and a single soft drink costs $.. 6. Let the cost per package of bacon. Let the cost of a carton of eggs. Set up a sstem of equations for the problem: Multipl each side of the first equation b and each side of the second equation b and solve b elimination: Substitute and solve for : (.9) A package of bacon costs $.9 and a carton of eggs cost $.9. The refund for packages of bacon and cartons of eggs will be ($.9) + ($.9) $ Copright Pearson Education, Inc. Publishing as Prentice Hall

14 Section.: Sstems of Linear Equations: Substitution and Elimination 66. Let Pamela s average speed in still water. Let the speed of the current. Rate Time Distance Downstream + Upstream Set up a sstem of equations for the problem: ( + ) ( ) Multipl each side of the first equation b, multipl each side of the second equation b, and add the result to eliminate : Pamela's average speed is miles per hour and the speed of the current is mile per hour. 68. Let the # of units of powder. Let the # of units of powder. Setting up the equations and solving b substitution:.+. vitamin B.+. vitamin E Multipling each equation b ields + 6+ Subtracting the bottom equation from the top equation ields + ( 6+ ) ( ) So units of powder should be mied with units of powder. 67. Let the # of mg of compound. Let the # of mg of compound. Setting up the equations and solving b substitution:.+. vitamin C.+. vitamin D Multipling each equation b ields Subtracting the bottom equation from the top equation ields ( ) ( ) So mg of compound should be mied with 7 mg of compound. 69. a + b+ c At (, ) the equation becomes: a( ) + b( ) + c a b+ c At (, ) the equation becomes: a() + b() + c a+ b+ c At (, ) the equation becomes: a() + b() + c c The sstem of equations is: a b+ c a+ b+ c c Substitute c into the first and second equations and simplif: a b+ a+ b+ a b a+ b a b+ Solve the first result for a, substitute into the second result and solve: 87 Copright Pearson Education, Inc. Publishing as Prentice Hall

15 Chapter : Sstems of Equations and Inequalities 7. ( b+ ) + b b+ + b 6b b a + The solution is a, b, c. The equation is +. a + b+ c At (, ) the equation becomes: a( ) + b( ) + c a b+ c At (, ) the equation becomes: a() + b() + c a+ b+ c At (, ) the equation becomes: a() + b() + c a+ b+ c The sstem of equations is: a b+ c a+ b+ c a+ b+ c Multipl the first equation b and add to the second equation; multipl the first equation b and add to the third equation to eliminate c: a + b c a+ b c a+ b+ c a+ b+ c a+ b 6 b a+ b b Substitute and solve: a + ( ) a c ab ( ) 6 The solution is a, b, c 6. The equation is Y r.6y + 6r 9 Multipl the first equation b, the add the result to the second equation to eliminate Y..6Y + r.6y + 6r 9 r 66 r.6 Substitute this result into the first equation to find Y..6Y (.6).6Y.6Y Y 9 The equilibrium level of income and interest rates is $9 million and 6%..Y r.y + 8r Multipl the first equation b, the add the result to the second equation to eliminate Y..Y + r. Y + 8r 8r 9 r. Substitute this result into the first equation to find Y..Y (.).Y.Y 6 Y The equilibrium level of income and interest rates is $ million and %. I I+ I I I I 7I Substitute the epression for I into the second and third equations and simplif: I ( I+ I) 8I I ( I+ I) 7I I I Multipl both sides of the first result b and multipl both sides of the second result b 8 to eliminate I : 88 Copright Pearson Education, Inc. Publishing as Prentice Hall

16 Section.: Sstems of Linear Equations: Substitution and Elimination 7. I I I + 96I 8 7I I 7 Substituting and solving for the other variables: 8I 7 7 8I 7 8 8I 7 I 7 6 I The solution is I, I 6, I I I+ I 8 I + 6I 8I + 6I Substitute the epression for I into the second equation and simplif: 8 ( I+ I) + 6I 8 I+ I 8I + 6I 8I 6I I + I 8 Multipl both sides of the first result b and add to the second result to eliminate I : 8I I 6 8 I 6 I 6I 6 I 6 Substituting and solving for the other variables: 6 I I + 8 I I 6 7 I I+ I + The solution is I, I 6 7, I. 7. Let the number of orchestra seats. Let the number of main seats. Let z the number of balcon seats. Since the total number of seats is, + + z. Since the total revenue is $7, if all seats are sold, + + z 7,. If onl half of the orchestra seats are sold, the revenue is $,6. So,, z. Thus, we have the following sstem: + + z + + z 7, + + z,6 Multipl each side of the first equation b and add to the second equation to eliminate z; multipl each side of the third equation b and add to the second equation to eliminate z: z, + + z 7, z 7, z, 6 Substituting and solving for the other variables: () z + z z 9 There are orchestra seats, main seats, and 9 balcon seats. 89 Copright Pearson Education, Inc. Publishing as Prentice Hall

17 Chapter : Sstems of Equations and Inequalities 76. Let the number of adult tickets. Let the number of child tickets. Let z the number of senior citizen tickets. Since the total number of tickets is, + + z. Since the total revenue is $, z. Twice as man children's tickets as adult tickets are sold. So,. Thus, we have the following sstem: + + z z Substitute for in the first two equations and simplif: + ( ) + z + z 8+.( ) + 6z 7+ 6z Multipl the first result b 6 and add to the second result to eliminate z: 8 6z 7+ 6z + z () () + z + z z 7 There were adults, children, and 7 senior citizens that bought tickets. 77. Let the number of servings of chicken. Let the number of servings of corn. Let z the number of servings of % milk. Protein equation: + + 9z 66 Carbohdrate equation: + 6+ z 9. Calcium equation: + + z 9 Multipl each side of the first equation b 6 and multipl each side of the second equation b and add them to eliminate ; multipl each side of the second equation b and multipl each side of the third equation b 8 and add to eliminate : 88 z z 8. 7 z z z 78 + z 687. Multipl each side of the first result b 9 and multipl each side of the second result b to eliminate : 7 99z, ,67z,7. 968z 9,6 z Substituting and solving for the other variables: 7 () (.) + + 9() The dietitian should serve. servings of chicken, serving of corn, and servings of % milk. 78. Let the amount in Treasur bills. Let the amount in Treasur bonds. Let z the amount in corporate bonds. Since the total investment is $,, + + z, Since the total income is to be $9,.+.7+.z 9 The investment in Treasur bills is to be $ more than the investment in corporate bonds. So, + z Substitute for in the first two equations and simplif: ( + z) + + z, + z 7, ( + z) + 7+ z 9, 7+ z, 9 Copright Pearson Education, Inc. Publishing as Prentice Hall

18 Section.: Sstems of Linear Equations: Substitution and Elimination Multipl each side of the first result b 7 and add to the second result to eliminate : 7 z 9, 7+ z, z, + z z 7, + () 7, +, 7, 7 Kell should invest $8 in Treasur bills, $7 in Treasur bonds, and $ in corporate bonds. 79. Let the price of hamburger. Let the price of order of fries. Let z the price of drink. We can construct the sstem z z.6 A sstem involving onl equations that contain or more unknowns cannot be solved uniquel. Multipl the first equation b and the second equation b, then add to eliminate : z. + + z.8 + z.7.7 z Substitute and solve for in terms of z:.7 ( z) + + z z.8 z+. z+ 6 Solutions of the sstem are:.7 z, z+. 6 Since we are given that.6 z.9, we choose values of z that give two-decimal-place values of and with.7. and.7.. The possible values of,, and z are shown in the table. z Let the price of hamburger. Let the price of order of fries. Let z the price of drink We can construct the sstem z z z.9 Subtract the second equation from the first equation to eliminate : z z.6 z. Multipl the third equation b and add it to the second equation to eliminate : z.6 96 z.8 z. Multipl the second result b and add it to the first result to eliminate : z. 8z. z 8 z.8 Substitute for z to find the other variables: (.8)....9 (.9) + + (.8) Therefore, one hamburger costs $.9, one order of fries costs $.9, and one drink costs $.8. 9 Copright Pearson Education, Inc. Publishing as Prentice Hall

19 Chapter : Sstems of Equations and Inequalities 8. Let Beth s time working alone. Let Bill s time working alone. Let z Edie s time working alone. We can use the following tables to organize our work: Beth Bill Edie Hours to do job z Part of job done in hour z In hours the complete entire job, so + + z + + z Bill Edie Hours to do job z Part of job done in hour z In hours the complete entire job, so +. z + z Beth Bill Edie Hours to do job z Part of job done in hour z With all working for hours and Beth and Bill working for an additional 8 hours, the complete entire job, so z + + z We have the sstem + + z + z + + z Subtract the second equation from the first equation: + + z + z Substitute into the third equation: + + z. + z Now consider the sstem consisting of the last result and the second original equation. Multipl the second original equation b and add it to the last result to eliminate : + z + z 8 z z Plugging z to find : + z + Working alone, it would take Beth hours, Bill hours, and Edie hours to complete the job Answers will var. 9 Copright Pearson Education, Inc. Publishing as Prentice Hall

20 Section.: Sstems of Linear Equations: Matrices Section.. matri. augmented. third; fifth. True. Writing the augmented matri for the sstem of equations: Writing the augmented matri for the sstem of equations: Write the sstem in standard form and then write the augmented matri for the sstem of equations: Write the sstem in standard form and then write the augmented matri for the sstem of equations: Writing the augmented matri for the sstem of equations: Writing the augmented matri for the sstem of equations: +. Writing the augmented matri for the sstem of equations: + z z. Writing the augmented matri for the sstem of equations: z + z. Writing the augmented matri for the sstem of equations: + z + z. + z z+ + z Write the sstem in standard form and then write the augmented matri for the sstem of equations: + z z + z. Writing the augmented matri for the sstem of equations: z + + z + + z 6. Writing the augmented matri for the sstem of equations: + z w + z+ w z w 9 Copright Pearson Education, Inc. Publishing as Prentice Hall

21 Chapter : Sstems of Equations and Inequalities R r + r () + ( ) ( ) + 9 R r + r () + ( ) ( ) + z z z 6 R r + r () + ( ) () + 6 () R r + r () ( ) + () + () z z 6 + z 6 R r + r 6 () ( ) () ( ) R r + r () ( ) () + ( ) z 6 + z z 6 R r + r () + ( ) () + ( 6) Copright Pearson Education, Inc. Publishing as Prentice Hall

22 Section.: Sstems of Linear Equations: Matrices.. R r + r () ( ) 6 () + ( 6) z z z 6 R 6r + r () + 6 6( ) 6( ) + 6 6( 6) a. b. R r+ r z 6 + 6z z 6. R r + r () + ( ) ( ) + (6) + ( ) z 6 + z z 6 R r + r () + ( ) () (6) R r + r ( ) + ( 6) R r + r 6 6 () + ( ) (6) + ( ) Consistent;,, or using ordered pairs (, ). Consistent;,, or using ordered pairs (,). 9 Copright Pearson Education, Inc. Publishing as Prentice Hall

23 Chapter : Sstems of Equations and Inequalities Inconsistent Inconsistent + z z Consistent; z + z z is an real number or {( z,, ) z, + z, z is an real number} + z + z Consistent; z z z is an real number or {( z,, ) z, z, z is an real number} + + Consistent; is an real number {(,,, ),, or, is an real number} 96 Copright Pearson Education, Inc. Publishing as Prentice Hall Consistent; is an real number {(,,, ),,, or is an real number} Consistent;, are an real numbers {(,,, ),, or and are an real numbers} + Consistent; is an real number {(,,, ),,, or is an real number} + + Consistent; is an real number {(,,, ),, or, is an real number}

24 Section.: Sstems of Linear Equations: Matrices Consistent; or (,,, ) + 8 Write the augmented matri: 8 8 ( R r+ r) 8 ( R r ) 6 ( R r + r) The solution is 6, or using ordered pairs (6, ). + + Write the augmented matri: R r+ r ( R r) ( R r + r) The solution is, or using ordered pairs (, ). + Write the augmented matri: ( ).. ( R ) r ( R r+ r ) 8 6 ( R ) r 8 ( R r + r) The solution is, or using ordered pairs, Write the augmented matri: 8 The solution is pairs,. ( R ) r Write the augmented matri: 8 This is a dependent sstem. + ( R r+ r) ( R ) r ( R r + r ), or using ordered ( R r+ r) The solution is, is an real number or {(, ), is an real number} 97 Copright Pearson Education, Inc. Publishing as Prentice Hall

25 Chapter : Sstems of Equations and Inequalities Write the augmented matri: 7 7 ( R r ) R 9r + r This is a dependent sstem. 7 7 ( ) The solution is 7, is an real number or {(, ) 7, is an real number} + 6 Write the augmented matri: 6 ( R r ) + ( R r ) + The solution is, or,. ( R r r ) ( R r r) + 8 Write the augmented matri: ( R r) 8 8 R r+ r ( R r ) R r + r The solution is, or (, ). ( ) ( ) Write the augmented matri: ( R r ) R r + r 6 ( R r ) ( R r + r) The solution is ( ), or,. + ( R r ) + ( R r + r) The solution is, or,. 6 z 6 + z Write the augmented matri: ( R r ) ( R r r ) ( R r r ) 98 Copright Pearson Education, Inc. Publishing as Prentice Hall

26 Section.: Sstems of Linear Equations: Matrices 8. 8 R r + r R r + r 8 ( R r ) 8 R r + r R r r + The solution is 8,, z or (8,, ). + + z z Write the augmented matri: ( R r) ( R r+ r) ( R r ) R r + r R r r + ( R r ) R r + r R r + r The solution is,, z or (,,) z z + z Write the augmented matri: 7 7 R r+ r R r+ r ( R r) R r + r R r + r ( R r) R r + r R r + r The solution is,, z or (,,). + z + + z 7 z 7 Write the augmented matri: ( R r) R r+ r 7 R r+ r 7 99 Copright Pearson Education, Inc. Publishing as Prentice Hall

27 Chapter : Sstems of Equations and Inequalities 7 7 ( R r) R 7 r + r R r + r ( R r) 6 7 R 7 r 6 + r R r + r 6 7 The solution is,, z or 6 7,,.. z + + z z Write the augmented matri: Interchange r and r R r+ r R r+ r 6 ( R r + r) There is no solution. The sstem is inconsistent.. z + + z + Write the augmented matri: ( R r) R r+ r R r + r ( R r + r) There is no solution. The sstem is inconsistent z + z 7z Write the augmented matri: 7 ( R r) 7 R r+ r R r+ r R r + r R r + r The matri in the last step represents the sstem z z z or, equivalentl, z The solution is z, z, z is an real number or {( z,, ) z, z, z is an real number}. Copright Pearson Education, Inc. Publishing as Prentice Hall

28 Section.: Sstems of Linear Equations: Matrices. z + + z + + z Write the augmented matri: Interchange and r r R r+ r 7 R r r 7 + R 7 r + r R r 6 7 ( R r + r) The matri in the last step represents the sstem 6 + z 7 + z or, equivalentl, 6 z + 7 z The solution is z+, z+, 6 z is an real number or ( z,, ) z+, 7 z+, z is an real number z 6 + z + 7z Write the augmented matri: 6 7 ( R r ) 7 R r+ r R r r R r + r R r + r 9 There is no solution. The sstem is inconsistent. + z z + z Write the augmented matri: ( R r) 8 R 7r+ r 8 R r+ r 8 ( R r + r) 9 There is no solution. The sstem is inconsistent. Copright Pearson Education, Inc. Publishing as Prentice Hall

29 Chapter : Sstems of Equations and Inequalities z 6 + z + z Write the augmented matri: 6 6 R r+ r R r+ r 8 6 ( R r ) 8 7 R r + r 6 R r + r 7 ( R r ) R r + r R r + r The solution is,, z, or (,, ). + z + z + z Write the augmented matri: R r+ r 7 R r r ( R r) R r + r 7 R 6r + r 7 ( R r) R r + r R r + r The solution is,, z or (,, ). + z + z 7 + z Write the augmented matri: 7 R r+ r 8 R r+ r 6 6 ( R r ) R r + r 8 8 R 6r + r ( R r ) 8 8 R r + r R r 8 + r The solution is,, z or,,. Copright Pearson Education, Inc. Publishing as Prentice Hall

30 Section.: Sstems of Linear Equations: Matrices z 8 + z + + 6z Write the augmented matri: R r+ r R r + r 8 6 ( R r ) R r + r R r + r ( R r 8 ) 9 9 R 8 r + r R r + r 9 The solution is + z + z 8 + Write the augmented matri: 8 9 R 8 8 8,, z or,, 9 9. ( r) 6. 9 R r+ r 9 6 R r+ r 9 9 ( R r ) 6 9 R r + r R r + r ( R r ) ( R r + r) The solution is,, z or,,. + + z z Write the augmented matri: 8 R r+ r R r + r Interchange r and r ( R r r) + Copright Pearson Education, Inc. Publishing as Prentice Hall

31 Chapter : Sstems of Equations and Inequalities 6. ( ) R r ( ) R r r ( R r r) The solution is,, z or,,. + + z+ w + z + + z w 6 z+ w Write the augmented matri: 6 R r+ r 8 R r+ r 6 R r r + 6 Interchange 8 r and r 6 ( R r) 8 R r + r 6 R r + r R r + r 6. 6 ( R r) R r + r R r + r R r r ( R r) R r + r R r + r The solution is,, z, w or (,,, ). + + z+ w + + z + + z w 6 + z+ w Write the augmented matri: 6 R r+ r R r+ r R r r + 7 Interchange r and r 7 6 R r + r R r + r R r r + 7 Copright Pearson Education, Inc. Publishing as Prentice Hall

32 Section.: Sstems of Linear Equations: Matrices 6. 6 ( R r r ) 6 R r R r Write the matri as the corresponding sstem: + z+ w 6 z w z+ w w Substitute and solve: z + () z + z () + () + () The solution is,, z, w or (,,, ). + + z + z + + z Write the augmented matri: R r+ r R r + r ( R r + r) The matri in the last step represents the sstem z Substitute and solve: + () + z z The solution is, z, is an real number or {( z,, ), z, is an real number}. + z + z 6 + z Write the augmented matri: 6 R r+ r R r+ r ( R r + r) There is no solution. The sstem is inconsistent. + z + z Write the augmented matri: ( R r+ r) ( R r ) ( R r + r) The matri in the last step represents the sstem or, equivalentl, z z Thus, the solution is, z, z is an real number or {( z,, ), z, z is an real number}. Copright Pearson Education, Inc. Publishing as Prentice Hall

33 Chapter : Sstems of Equations and Inequalities z + + z Write the augmented matri: interchange r and r ( R r+ r) 6 ( R r ) ( R r + r) The matri in the last step represents the sstem z + z or, equivalentl, + z z Thus, the solution is: + z, z, z is an real number or ( z,, ) + z, z, z is an real number. + z z + + z + + z Write the augmented matri: interchange r and r R r r + R r+ r R r r + 7. interchange r and r 7 7 ( R r + r) R r + r 8 9 R r + r ( R r 8 ) 8 The matri in the last step represents the sstem 8z 7 7z 7 9 z 8 Substitute and solve: Thus, the solution is,, z or ,, z z + z Write the augmented matri: 6 Copright Pearson Education, Inc. Publishing as Prentice Hall

34 Section.: Sstems of Linear Equations: Matrices 7. R r+ r 6 R r + r R r+ r interchange r and r 6 R r + r R r + r R r + r R r + r R r + r There is no solution. The sstem is inconsistent. + + z w + z + w Write the augmented matri: interchange r and r ( R r+ r) 7 8 The matri in the last step represents the sstem + z+ w 7z w 8 The second equation ields 7z w8 7z+ w8 7 8 z+ w The first equation ields + z+ w + zw Substituting for : z+ wzw 7 z w+ 7 7 Thus, the solution is z w+, 7 8 z+ w, z and w are an real numbers or ( zw,,, ) z w+, z+ w, z and w are an real numbers. Copright Pearson Education, Inc. Publishing as Prentice Hall z w z+ w Write the augmented matri: ( R r ) ( R r+ r) ( R r) ( R r + r) R r + r 7 R r + r The matri in the last step represents the sstem + w w + w 7 or, equivalentl, 7 w z+ w z w The solution is w, 7 w, z w, w is an real number or {( zw,,, ) w, 7 w, z w, w is an real number}

35 Chapter : Sstems of Equations and Inequalities 7. Each of the points must satisf the equation a + b+ c. (, ) : a+ b+ c (, 7): 7 a b+ c (, ) : a+ b+ c Set up a matri and solve: 7 R r+ r 6 R r+ r ( R r 6 ) R r + r R r + r 6 ( ) R r R r + r R r + r The solution is a, b, c ; so the equation is Each of the points must satisf the equation a + b+ c. (, ) : a+ b+ c (, ) : 9a+ b+ c (,) : a b+ c Set up a matri and solve: 9 R 9r+ r R r+ r 6 8 R r 6 R r + r R r + r R r R r + r R r r + The solution is a, b, c ; so the equation is Each of the points must satisf the equation f ( ) a + b + c + d. f( ) : 7a+ 9b c+ d f( ) : a+ b c+ d f() : a+ b+ c+ d f() : 8a+ b+ c+ d Set up a matri and solve: Interchange 7 9 r and r 8 R r+ r R 7 r+ r 6 8 R 8r r ( R r) R r + r R 6r + r 8 R r r Copright Pearson Education, Inc. Publishing as Prentice Hall

36 Section.: Sstems of Linear Equations: Matrices ( R r ) 6 R r + r R 6r + r ( ) R r R r + r R r + r R r + r The solution is a, b, c, d ; so the equation is f( ) Each of the points must satisf the equation f ( ) a + b + c + d. f( ) : 8a+ b c+ d f( ) : a+ b c+ d f() : a+ b+ c+ d f() : 7a+ 9b+ c+ d Set up a matri and solve: Interchange 8 r and r 7 9 R r+ r 8 R 8r+ r 6 9 R 7r r ( R r) R r + r R r + r 6 8 R 8r r ( R r 6 ) 8 8 R r + r R r + r ( R r ) 6 R r + r R r + r R r + r 6 The solution is a, b, c, d 6 ; so the equation is f( ) Let the number of servings of salmon steak. Let the number of servings of baked eggs. Let z the number of servings of acorn squash. Protein equation: + + z 78 Carbohdrate equation: + + z 9 Vitamin A equation: + + z 7 Set up a matri and solve: Interchange 9 r and r ( R r ) R r+ r R r+ r Copright Pearson Education, Inc. Publishing as Prentice Hall

37 Chapter : Sstems of Equations and Inequalities ( R r 7 ) Substitute z and solve: 98 9() () + 6() ( R r r) The dietitian should serve. servings of salmon steak, servings of baked eggs, and serving of acorn squash. 78. Let the number of servings of pork chops. Let the number of servings of corn on the cob. Let z the number of servings of % milk. Protein equation: + + 9z 7 Carbohdrate equation: 6+ z 8 Calcium equation: + + z 6 Set up a matri and solve: Interchange 6 8 r and r R 9 r+ r 6 8 R r R r + r 6 8 R r + r ( R r 69 ) R 67 r 6 + r R r 6 + r The dietitian should provide serving of pork chops, servings of corn on the cob, and servings of % milk. 79. Let the amount invested in Treasur bills. Let the amount invested in Treasur bonds. Let z the amount invested in corporate bonds. Total investment equation: + + z, Annual income equation: z 68 Condition on investment equation: z. z Set up a matri and solve:, , R.6r+ r.. 8 R, r+ r, 8 ( R r), R r + r 8 R r + r 8 ( R r) R r + r R r + r Carletta should invest $ in Treasur bills, $ in Treasur bonds, and $ in corporate bonds. 8. Let the fied deliver charge; let the cost of each tree, and let z the hourl labor charge. st subdivision: z 7 nd subdivision: + + z 9 rd subdivision: + + z 898 Set up a matri and solve: Copright Pearson Education, Inc. Publishing as Prentice Hall

38 Section.: Sstems of Linear Equations: Matrices 66 7 R rr 7 6 R r r 66 7 R r.8. R r R rr R r r.8. ( R r.6 ).7 R r+ r 9.9 R r.8r.7 The deliver charge is $ per job, the cost for each tree is $9.9, and the hourl labor charge is $ Let the number of Deltas produced. Let the number of Betas produced. Let z the number of Sigmas produced. Painting equation: z Dring equation: + + z 69 Polishing equation: + + z Set up a matri and solve: ( R r + r) R r+ r R r+ r ( R r ) 8 R rr R r r 8 ( R r) 8 R r + r R r + r The compan should produce 8 Deltas, Betas, and Sigmas. 8. Let the number of cases of orange juice produced; let the number of cases of grapefruit juice produced; and let z the number of cases of tomato juice produced. Sterilizing equation: 9+ + z 98 Filling equation: 6+ + z 6 Labeling equation: + + z 8 Set up a matri and solve: Interchange 6 6 and 9 98 r r 8 R 6r+ r 8 8 R 9r r ( R r 8 ) 8 R r + r R 8r + r 6 ( R r ) 6 R r + r R r r + The compan should prepare 6 cases of orange juice, cases of grapefruit juice, and cases of tomato juice. Copright Pearson Education, Inc. Publishing as Prentice Hall

39 Chapter : Sstems of Equations and Inequalities 8. Rewrite the sstem to set up the matri and solve: + 8 I I 8 I I + I+ I 8 I + I I+ I I + I I II I 8 8 Interchange r and r 8 R r R r+ r R r r Interchange 6 8 r and r 8 R r 6 8 R r + r 8 8 ( R r ) R r + r R 6 r + r The solution is I, I, I, 8 I. 8. Rewrite the sstem to set up the matri and solve: I I + I II I 6I I 6I I 6I 6I + 6I 6I R 6r+ r 6 9 R 6r+ r 6 6 ( R r 6 ) 6 6 R r + r R r + r ( R r ). R r + r. R r + r The solution is I., I., I. 8. Let the amount invested in Treasur bills. Let the amount invested in corporate bonds. Let z the amount invested in junk bonds. a. Total investment equation: + + z, Annual income equation: z Set up a matri and solve:,.7.9., 7 9,, 6,,, ( R r ) ( R r 7r ) ( R r) Copright Pearson Education, Inc. Publishing as Prentice Hall

40 Section.: Sstems of Linear Equations: Matrices,, ( R rr) The matri in the last step represents the z, sstem + z, Therefore the solution is, + z,, z, z is an real number. Possible investment strategies: Amount Invested At 7% 9% %,, 8, 6,,,, b. Total investment equation: + + z, Annual income equation: z Set up a matri and solve:,.7.9., 7 9, ( R r ),,,, ( R r 7r ) ( R r), ( R rr), The matri in the last step represents the z, sstem + z, Thus, the solution is z+,, z+,, z is an real number. Possible investment strategies: Amount Invested At 7% 9% %,,, 8 6, 8,7 6 c. Total investment equation: + + z, Annual income equation: z Set up a matri and solve:,.7.9., ( R r) 7 9,, ( R r 7r),, ( R r ), ( R rr) The matri in the last step represents the z, sstem + z Thus, the solution is z+,, z, z is an real number. However, and z cannot be negative. From z, we must have z. One possible investment strateg Amount Invested At 7% 9% %, This will ield ($,)(.7) $, which is more than the required income. d. Answers will var. Copright Pearson Education, Inc. Publishing as Prentice Hall

41 Chapter : Sstems of Equations and Inequalities 86. Let the amount invested in Treasur bills. Let the amount invested in corporate bonds. Let z the amount invested in junk bonds. Let I income Total investment equation: + + z, Annual income equation: z I Set up a matri and solve:,.7.9. I, ( R r) 7 9 I, ( R r 7r) I 7,, ( R r ) I 87,, I ( R rr) I 87, The matri in the last step represents the sstem z, I + z I 87, Thus, the solution is, I + z, I 87, z, z is an real number. a. I, I + z, ( ) + z 7, + z I 87, z ( ) 87, z, z z is an real number. Since and z cannot be negative, we must have z. Investing all of the mone at 7% ields $7, which is more than the $ needed. b. I, T + z, ( ) + z, + z I 87, z ( ) 87, z, z z is an real number. Possible investment strategies: Amount Invested At 7% 9% %,,, 6 8,7 6 c. I, T + z, ( ) + z, + z I 87, z ( ) 87, z 7, z z is an real number. Possible investment strategies: Amount invested at 7% 9% %,,,, 6 8,7 d. Answers will var. 87. Let the amount of supplement. Let the amount of supplement. Let z the amount of supplement..+.+.z Vitamin C.+.+.z Vitamin D Multipling each equation b ields + + z + + z Set up a matri and solve: ( R r ) 7 ( R r ) 8 ( R r r ) Copright Pearson Education, Inc. Publishing as Prentice Hall

42 Section.: Sstems of Linear Equations: Determinants 7 ( R r r) 7 8 The matri in the last step represents the sstem 7 + z z 7 8 Therefore the solution is 7 z, 7 + z, z is an real number. 8 Possible combinations: Supplement Supplement Supplement mg 7mg mg 6mg 76mg 8mg mg 77mg 6mg 8mg 78mg mg 88. Let the amount of powder. Let the amount of powder. Let z the amount of powder..+.+.z Vitamin B.+.+.z Vitamin E Multipling each equation b ields + + z + + z Set up a matri and solve: ( R r r ) ( R r+ r). 6 The matri in the last step represents the sstem +.z 6.z 6 Thus, the solution is.z,.z, z is an real number. Possible combinations: Powder Powder Powder units units units units units 8 units units units 6 units units units units Answers will var. Section.. ad bc.. False; If adbc, the the det.. False; The solution cannot be determined.. False; See Thm () 6. False; See Thm () () ( )() () ( ) () ( ) 6 + () ( )() + + ( ) [ ] + [ () ( ) ] ( ) () ( ) () ( 8) ( 7) + () Copright Pearson Education, Inc. Publishing as Prentice Hall

43 Chapter : Sstems of Equations and Inequalities ( ) [ ] [ ] [ 6() 8() ] () ( ) 6() 8( ) () (8) () ( ) + [ ] [ ] + [ 6( ) ( ) ] () ( ) + 6() () ( ) + () + ( 7) ( 9) [ ] [ ] + [ ( ) 8() ] () ( )() + 9 () 8() () + 9() + ( ) D 8 D 8 8 D 8 Find the solutions b Cramer's Rule: D D 6 D D The solution is (6, ) D D 6 D Find the solutions b Cramer's Rule: D D D D The solution is,. + D + 7 D 9 + D 6 6 Find the solutions b Cramer's Rule: D D D 7 D 7 The solution is (, ). + 8 D 69 D ( ) 9 8 D Find the solutions b Cramer's Rule: D 9 D 8 D 9 D 9 The solution is (,). 6 Copright Pearson Education, Inc. Publishing as Prentice Hall

44 Section.: Sstems of Linear Equations: Determinants D 6 6 D D 8 8 D Find the solutions b Cramer's Rule: D 8 D 8 D 6 D 6 The solution is (8, ). D D Find the solutions b Cramer's Rule: D D 66 D D The solution is (, ).. + D 8 8 D 6 ( ) 6 D 6 6 Find the solutions b Cramer's Rule: D 6 D 6 D 8 D D ( ) 6 Since D, Cramer's Rule does not appl D Since D, Cramer's Rule does not appl. The solution is, D ( ) D 6 96 ( 7) 68 D 6 8 Find the solutions b Cramer's Rule: D 68 D 8 D D The solution is (, ). D + 6 D + 8 D 6+ 6 Find the solutions b Cramer's Rule: D 8 D D 6 D 6 The solution is,. 7 Copright Pearson Education, Inc. Publishing as Prentice Hall

45 Chapter : Sstems of Equations and Inequalities D 6 6 D D 8 8 Find the solutions b Cramer's Rule: D D D 6 D 6 The solution is, D 6 D 6 6 D 6 Find the solutions b Cramer's Rule: D D D D The solution is, D ( ) D ( ) D ( ) Find the solutions b Cramer's Rule: D D D D The solution is, D D 8 8 D ( ) 6 8 Find the solutions b Cramer's Rule: D D 6 D D The solution is (, ) D ( ) D ( 8) 8 D Find the solutions b Cramer's Rule: D 8 D D D The solution is,. + D ( 7) 9 D ( ) D 6 8 Find the solutions b Cramer's Rule: D D 8 D 9 D 9 The solution is,. 8 Copright Pearson Education, Inc. Publishing as Prentice Hall

46 Section.: Sstems of Linear Equations: Determinants. + D + D + D + Find the solutions b Cramer's Rule: D D D D The solution is,. Dz ( 8 + ) ( + ) + 6(9 + ) Find the solutions b Cramer's Rule: D D 9 D D Dz 6 z D The solution is (,, ).. + z 6 + z + z D D D + ( ) ( ) ( 6 ) (9 + ) ( ) 6( ) ( ) ( + 8) z + z + z D D D ( ) + (6 ) + ( ) + (+ ) + 7 ( ) + (6 ) + ( 8) + ( + 6) ( ) ( ) 6( 6 ) ( + ) ( ) + ( 8) + ( ) + ( + 7) Copright Pearson Education, Inc. Publishing as Prentice Hall

47 Chapter : Sstems of Equations and Inequalities D z D z 7. ( ) + ( ) ( 6 + ) + ( + 7) ( + ) Find the solutions b Cramer's Rule: D D D D Dz z D The solution is (,, ). + z + z 7 + z D D D + ( ) ( ) ( 6 + ) ( 8) ( ) 7 ( ) () ( + 6) ( ) + ( ) ( ) + ( 6 + ) (8 ) ( ) ( 6 + ) (8 ) ( 8) + + Find the solutions b Cramer's Rule: D 66 D D D Dz z D The solution is,,. + z 8 + z + + 6z D 6 D D + ( ) 6 6 ( 6 ) (8 ) ( + ) ( ) 6 6 8( 6) (7) (+ ) ( 8) + ( ) 6 6 (7 ) + 8(8 ) ( ) Copright Pearson Education, Inc. Publishing as Prentice Hall

48 Section.: Sstems of Linear Equations: Determinants D z z + z + z ( 8) ( ) ( ) 8( + ) Find the solutions b Cramer's Rule: D D 6 8 D 8 D 8 Dz 9 z D The solution is,, 9. + z + z + 6z D 6 ( ) (6 8) + (8 + ) + ( ) + Since D, Cramer's Rule does not appl. + z + + z D ( ) + ( 8 ) + ( ) + (6 + ) 8 + Since D, Cramer's Rule does not appl.. D D D Dz + ( ) ( ) ( 6 + ) ( 8) [B Theorem ()] [B Theorem ()] [B Theorem ()] Find the solutions b Cramer's Rule: D D D D Dz z D The solution is (,, ). + z + z + + 6z D 6 D + ( ) 6 6 ( 6 ) (8 ) ( + ) 96 8 [B Theorem ()] 6 Copright Pearson Education, Inc. Publishing as Prentice Hall

49 Chapter : Sstems of Equations and Inequalities.. D D z [B Theorem ()] 6 [B Theorem ()] Find the solutions b Cramer's Rule: D D D 8 D 8 Dz z D 8 The solution is (,, ). + z + z + 6z D 6 ( ) (6 8) + (8+ ) + ( ) + Since D, Cramer's Rule does not appl. + z + + z D ( ) + ( 8 ) + ( ) + (6 + ) 8 + Since D, Cramer's Rule does not appl. z. u v w B Theorem (), the value of a determinant changes sign if an two rows are interchanged. Thus, u v w. z z. u v w B Theorem (), if an row of a determinant is multiplied b a nonzero number k, the value of the determinant is also changed b a factor of k. z z Thus, u v w u v w () 8. 6 z. Let u v w. z z 6 9 [Theorem ()] u v w u v w z ( ) u v w [Theorem ()] () z 6. Let u v w [Theorem ()] u v z w z ( R r + r) u v w u v w z ( ) [Theorem ()] u v w z ( )( ) u v w [Theorem ()] z u v w Copright Pearson Education, Inc. Publishing as Prentice Hall

50 Section.: Sstems of Linear Equations: Determinants z 7. Let u v w 6 z9 u v w 6 z9 [Theorem ()] u v w 6 z9 ( ) [Theorem ()] u v w 6 z9 ( )( ) u v w [Theorem ()] ( )( ) z u v w ( )( )() 8 z 8. Let u v w z z u v w u u v w z 9. Let u v w [Theorem ()] ( R r + r) [Theorem ()] ( C c + c ) z u v w z [Theorem ()] u v w z ( ) [Theorem ()] u v w z ( )( ) u v w [Theorem ()] z ( )( ) u v w ( )( )() 8 z. Let u v w z+ 9 u v w z u v w z u v w z [Theorem ()] ( R r + r ) [Theorem ()] ( R r + r) [Theorem ()] ( R r + r ) u v w [Theorem ()] (). Solve for :. Solve for : ( )( + ) or + or Copright Pearson Education, Inc. Publishing as Prentice Hall

51 Chapter : Sstems of Equations and Inequalities. Solve for : 6. Solve for : ( ) ( ) ( ) +. Solve for : + + ( ) ( ) ( ) Solve for : ( ) ( ) ( ) ( ) + 9 or ( ) ( ) ( ) + + ( ) + or 7. Epanding the determinant: ( ) ( ) ( ) + ( ) + ( ) ( ) + ( ) ( ) ( ) + ( ) ( ) ( ) ( ) + ( ) ( ) ( ) + + ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) This is the -point form of the equation for a line. Copright Pearson Education, Inc. Publishing as Prentice Hall

52 Section.: Sstems of Linear Equations: Determinants 8. An point (, ) on the line containing (, ) and (, ) satisfies: If the point (, ) is on the line containing (, ) and (, ) [the points are collinear], then. Conversel, if, then (, ) is on the line containing (, ) and (, ), and the points are collinear. 9. If the vertices of a triangle are (, ), (, ), and (6, ), then: 6 D + 6 [ ( ) ( ) + 6( ) ] [ ( ) ( ) + 6() ] [ ] The area of the triangle is square units. 6. Epanding the determinant: z z z z z z + ( z) ( z ) + ( zz ) ( z) ( z)( + z) + z( z) ( z) z z + ( z) [ ( ) z( ) ] ( z)( )( z) 6. If a, then b and c since b s ad bc, and the sstem is. c + d t s The solution of the sstem is, b s td t d( b ) tbsd. Using Cramer s c c bc b Rule, we get D bc c d s b D sd tb t d s D sc sc c t D ds tb td sd and D bc bc D sc s, which is the solution. Note D bc b that these solutions agree if d. If b, then a and d since a s ad bc, and the sstem is. c + d t s The solution of the sstem is, a tc atcs. Using Cramer s Rule, we d ad a get D ad c d, s D sd t d, and Copright Pearson Education, Inc. Publishing as Prentice Hall