Systems of Linear Equations

Transcription

1 Sstems of Linear Equations. Solving Sstems of Linear Equations b Graphing. Solving Sstems of Equations b Using the Substitution Method. Solving Sstems of Equations b Using the Addition Method. Applications of Sstems of Linear Equations in Two Variables. Sstems of Linear Equations in Three Variables and Applications.6 Solving Sstems of Linear Equations b Using Matrices.7 Determinants and Cramer s Rule In this chapter we solve sstems of linear equations in two and three variables. Some new terms are introduced in the first section of this chapter. Unscramble each word to find a ke word from this chapter. As a hint, there is a clue for each word. Complete the word scramble to familiarize ourself with the ke terms.. NNEDPNTDIEE (A sstem of two linear equations representing more than one line). NNCIEOSTINST (A sstem having no solution). LOUINSTO (An ordered pair that satisfies both equations in a sstem of two equations). INENSCOSTT (A sstem of equations that has one or more solutions). DNEEDPETN (A sstem of two linear equations that represents onl one line) 77

2 78 Chapter Sstems of Linear Equations Section. Concepts. Solutions to Sstems of Linear Equations. Dependent and Inconsistent Sstems of Linear Equations. Solving Sstems of Linear Equations b Graphing Solving Sstems of Linear Equations b Graphing. Solutions to Sstems of Linear Equations A linear equation in two variables has an infinite number of solutions that form a line in a rectangular coordinate sstem. Two or more linear equations form a sstem of linear equations. For eample: 0 A solution to a sstem of linear equations is an ordered pair that is a solution to each individual linear equation. Eample Determining Solutions to a Sstem of Linear Equations Determine whether the ordered pairs are solutions to the sstem. a., b. 0, 6 a. Substitute the ordered pair, into both equations: True True Because the ordered pair, is a solution to both equations, it is a solution to the sstem of equations. b. Substitute the ordered pair 0, 6 into both equations: True False Because the ordered pair 0, 6 is not a solution to the second equation, it is not a solution to the sstem of equations. Skill Practice Determine whether the ordered pairs are solutions to the sstem. 8 8 a. (, ) b. (, 0) Skill Practice Answers a. No b. Yes

3 Section. Solving Sstems of Linear Equations b Graphing 79 A solution to a sstem of two linear equations ma be interpreted graphicall as a point of intersection between the two lines. Notice that the lines intersect at, (Figure -). (, ) 6 Figure -. Dependent and Inconsistent Sstems of Linear Equations When two lines are drawn in a rectangular coordinate sstem, three geometric relationships are possible:. Two lines ma intersect at eactl one point.. Two lines ma intersect at no point. This occurs if the lines are parallel.. Two lines ma intersect at infinitel man points along the line. This occurs if the equations represent the same line (the lines are coinciding). If a sstem of linear equations has one or more solutions, the sstem is said to be a consistent sstem. If a linear equation has no solution, it is said to be an inconsistent sstem. If two equations represent the same line, then all points along the line are solutions to the sstem of equations. In such a case, the sstem is characterized as a dependent sstem. An independent sstem is one in which the two equations represent different lines. Solutions to Sstems of Linear Equations in Two Variables One unique solution No solution Infinitel man solutions One point of intersection Parallel lines Coinciding lines Sstem is consistent. Sstem is inconsistent. Sstem is consistent. Sstem is independent. Sstem is independent. Sstem is dependent.

4 80 Chapter Sstems of Linear Equations. Solving Sstems of Linear Equations b Graphing Eample Solving a Sstem of Linear Equations b Graphing Solve the sstem b graphing both linear equations and finding the point(s) of intersection. 6 To graph each equation, write the equation in slope-intercept form m b. First equation: Second equation: Slope: Slope: From their slope-intercept forms, we see that the lines have different slopes, indicating that the lines must intersect at eactl one point. Using the slope and -intercept we can graph the lines to find the point of intersection (Figure -). 6 Point of intersection (, ) Figure - The point, appears to be the point of intersection. This can be confirmed b substituting and into both equations. Skill Practice Answers. (, ) 9 True True The solution is,. Skill Practice. Solve b using the graphing method.

5 Section. Solving Sstems of Linear Equations b Graphing 8 TIP: In Eample, the lines could also have been graphed b using the - and -intercepts or b using a table of points. However, the advantage of writing the equations in slope-intercept form is that we can compare the slopes and -intercepts of each line.. If the slopes differ, the lines are different and nonparallel and must cross in eactl one point.. If the slopes are the same and the -intercepts are different, the lines are parallel and do not intersect.. If the slopes are the same and the -intercepts are the same, the two equations represent the same line. Eample Solve the sstem b graphing. Solving a Sstem of Linear Equations b Graphing The first equation 8 can be written as. This is an equation of a vertical line. To graph the second equation, write the equation in slopeintercept form. First equation: 8 Second equation: Figure - 8 (, 0) The graphs of the lines are shown in Figure -. The point of intersection is (, 0). This can be confirmed b substituting (, 0) into both equations. 8 8 True True The solution is (, 0). Skill Practice. Solve the sstem b graphing. Skill Practice Answers. (, )

6 8 Chapter Sstems of Linear Equations Eample Solving a Sstem of Equations b Graphing Solve the sstem b graphing Figure - To graph the line, write each equation in slope-intercept form. First equation: Second equation: Because the lines have the same slope but different -intercepts, the are parallel (Figure -). Two parallel lines do not intersect, which implies that the sstem has no solution. The sstem is inconsistent. Skill Practice. Solve the sstem b graphing. 0 Eample Solve the sstem b graphing. Solving a Sstem of Linear Equations b Graphing 8 Skill Practice Answers. No solution; inconsistent sstem Write the first equation in slope-intercept form. The second equation is alread in slope-intercept form. First equation: Second equation:

7 Section. Solving Sstems of Linear Equations b Graphing 8 Notice that the slope-intercept forms of the two lines are identical. Therefore, the equations represent the same line (Figure -). The sstem is dependent, and the solution to the sstem of equations is the set of all points on the line. Because not all the ordered pairs in the solution set can be listed, we can write the solution in set-builder notation. Furthermore, the equations 8 and represent the same line. Therefore, the solution set ma be written as, 0 6 or Skill Practice. Solve the sstem b graphing. Figure -, Calculator Connections The solution to a sstem of equations can be found b using either a Trace feature or an Intersect feature on a graphing calculator to find the point of intersection between two curves. For eample, consider the sstem 6 First graph the equations together on the same viewing window. Recall that to enter the equations into the calculator, the equations must be written with the -variable isolated. That is, be sure to solve for first. Isolate. 6 6 Skill Practice Answers B inspection of the graph, it appears that the solution is,. The Trace option on the calculator ma come close to, but ma not show the eact solution (Figure -6). However, an Intersect feature on a graphing calculator ma provide the eact solution (Figure -7). See our user s manual for further details.. {, 0 }; infinitel man solutions; dependent sstem

8 8 Chapter Sstems of Linear Equations Using Trace Using Intersect Figure -6 Figure -7 Section. Boost our GRADE at mathzone.com! Stud Skills Eercises Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos. Before ou proceed further in Chapter, make our test corrections for the Chapter test. See Eercise of Section. for instructions.. Define the ke terms. a. Sstem of linear equations b. Solution to a sstem of linear equations c. Consistent sstem d. Inconsistent sstem e. Dependent sstem f. Independent sstem Concept : Solutions to Sstems of Linear Equations For Eercises 8, determine which points are solutions to the given sstem ,,,, (, ) 0, 7, 0, 0, a, 9 b 7 0, 0, a,,, b ,,, 0, a, b 9,, 6, 0,, 0,,,, 9,

9 Section. Solving Sstems of Linear Equations b Graphing 8 Concept : Dependent and Inconsistent Sstems of Linear Equations For Eercises 9, the graph of a sstem of linear equations is given. a. Identif whether the sstem is consistent or inconsistent. b. Identif whether the sstem is dependent or independent. c. Identif the number of solutions to the sstem Concept : Solving Sstems of Linear Equations b Graphing For Eercises, solve the sstems of equations b graphing

10 86 Chapter Sstems of Linear Equations

11 Section. Solving Sstems of Linear Equations b Graphing For Eercises 6, identif each statement as true or false.. A consistent sstem is a sstem that alwas has a unique solution.. A dependent sstem is a sstem that has no solution.. If two lines coincide, the sstem is dependent. 6. If two lines are parallel, the sstem is independent. Graphing Calculator Eercises For Eercises 7, use a graphing calculator to graph each linear equation on the same viewing window. Use a Trace or Intersect feature to find the point(s) of intersection

12 v 88 Chapter Sstems of Linear Equations Section. Concepts. The Substitution Method. Solving Inconsistent Sstems and Dependent Sstems Solving Sstems of Equations b Using the Substitution Method. The Substitution Method Graphing a sstem of equations is one method to find the solution of the sstem. In this section and Section., we will present two algebraic methods to solve a sstem of equations. The first is called the substitution method. This technique is particularl important because it can be used to solve more advanced problems including nonlinear sstems of equations. The first step in the substitution process is to isolate one of the variables from one of the equations. Consider the sstem Solving the first equation for ields 6. Then, because is equal to 6, the epression 6 can replace in the second equation. This leaves the second equation in terms of onl. First equation: 6 6 Solve for. 6 Second equation: Substitute 6. 6 Solve for. 6 To find, substitute 6 back into the equation 6. The solution is (0, 6).

13 Section. Solving Sstems of Equations b Using the Substitution Method 89 Solving a Sstem of Equations b the Substitution Method. Isolate one of the variables from one equation.. Substitute the quantit found in step into the other equation.. Solve the resulting equation.. Substitute the value found in step back into the equation in step to find the value of the remaining variable.. Check the solution in both equations, and write the answer as an ordered pair. Eample Using the Substitution Method to Solve a Linear Equation Solve the sstem b using the substitution method. 9 9 Step : In the second equation, is alread isolated. a b 9 Step : Substitute the quantit for in the other equation. Step : Solve for. Now use the known value of to solve for the remaining variable. 6 9 Step : Substitute into the equation. Step : Check the ordered pair (, ) in each original equation True The solution is (, ). True

14 90 Chapter Sstems of Linear Equations Skill Practice. Solve b using the substitution method. 0 Eample Using the Substitution Method to Solve a Linear Sstem Solve the sstem b using the substitution method. The variable in the second equation is the easiest variable to isolate because its coefficient is Step : Solve the second equation for Step : Substitute the quantit 6 6 for in the other equation. Step : Solve for. Avoiding Mistakes: Do not substitute 6 6 into the same equation from which it came. This mistake will result in an identit: a b a b a b 6 6 Step : Substitute into the equation 6 6. Step : Check the ordered pair, in each original equation. The solution is,. Skill Practice Answers. (, )

15 Section. Solving Sstems of Equations b Using the Substitution Method 9 Skill Practice. Solve b the substitution method. 8. Solving Inconsistent Sstems and Dependent Sstems Eample Using the Substitution Method to Solve a Linear Sstem Solve the sstem b using the substitution method There is no solution. The sstem is inconsistent. Step : The variable is alread isolated. Step : Substitute the quantit into the other equation. Step : Solve for. The equation reduces to a contradiction, indicating that the sstem has no solution. The lines never intersect and must be parallel. The sstem is inconsistent. TIP: The answer to Eample can be verified b writing each equation in slope-intercept form and graphing the equations. Equation Equation Notice that the equations have the same slope, but different -intercepts; therefore, the lines must be parallel. There is no solution to this sstem of equations. 6 Skill Practice. Solve b the substitution method. 8 6 Skill Practice Answers. (, ). No solution; Inconsistent sstem

16 v 9 Chapter Sstems of Linear Equations Eample Solving a Dependent Sstem Solve b using the substitution method. 6 6 Step : Solve for one of the variables. 6 Step : Substitute the quantit for in the other equation. 6 6 Step : Solve for. 6 6 The sstem reduces to the identit 6 6. Therefore, the original two equations are equivalent, and the sstem is dependent. The solution consists of all points on the common line. Because the equations 6 and represent the same line, the solution ma be written as, 0 66 or, 0 6 Skill Practice. Solve the sstem b using substitution. 6 TIP: We can confirm the results of Eample b writing each equation in slope-intercept form. The slope-intercept forms are identical, indicating that the lines are the same. Skill Practice Answers. Infinitel man solutions;, 0 6 6; Dependent sstem 6 6 slope-intercept form Section. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos. Check our progress b answering these questions. Yes No Did ou have sufficient time to stud for the test on Chapter? If not, what could ou have done to create more time for studing?

17 Section. Solving Sstems of Equations b Using the Substitution Method 9 Yes No Did ou work all of the assigned homework problems in Chapter? Yes No If ou encountered difficult, did ou see our instructor or tutor for help? Yes No Have ou taken advantage of the tetbook supplements such as the Student Solutions Manual and MathZone? Review Eercises For Eercises, using the slope-intercept form of the lines, a. determine whether the sstem is consistent or inconsistent and b. determine whether the sstem is dependent or independent For Eercises 6 7, solve the sstem b graphing Concept : The Substitution Method For Eercises 8 7, solve b using the substitution method Describe the process of solving a sstem of linear equations b using substitution. Concept : Solving Inconsistent Sstems and Dependent Sstems For Eercises 9 6, solve the sstems

18 9 Chapter Sstems of Linear Equations When using the substitution method, eplain how to determine whether a sstem of linear equations is dependent. 8. When using the substitution method, eplain how to determine whether a sstem of linear equations is inconsistent. Mied Eercises For Eercises 9 0, solve the sstem b using the substitution method

19 Section. Solving Sstems of Equations b Using the Addition Method 9 Solving Sstems of Equations Section. b Using the Addition Method. The Addition Method The net method we present to solve sstems of linear equations is the addition method (sometimes called the elimination method). With the addition method, begin b writing both equations in standard form A B C. Then we create an equivalent sstem b multipling one or both equations b appropriate constants to create opposite coefficients on either the - or the -variable. Net the equations can be added to eliminate the variable having opposite coefficients. This process is demonstrated in Eample. Concepts. The Addition Method. Solving Inconsistent Sstems and Dependent Sstems Eample Solving a Sstem b the Addition Method Solve the sstem b using the addition method Multipl b Check the ordered pair (, ) in each original equation: Multipl the second equation b. This makes the coefficients of the -variables opposite. Now if the equations are added, the -variable will be eliminated. Solve for. Substitute back into one of the original equations and solve for The solution is (, ). TIP: Substituting into the other equation, 9, produces the same value for Skill Practice. Solve b the addition method. Skill Practice Answers. (, )

20 96 Chapter Sstems of Linear Equations The steps to solve a sstem of linear equations in two variables b the addition method is outlined in the following bo. Solving a Sstem of Equations b the Addition Method. Write both equations in standard form: A B C. Clear fractions or decimals (optional).. Multipl one or both equations b nonzero constants to create opposite coefficients for one of the variables.. Add the equations from step to eliminate one variable.. Solve for the remaining variable. 6. Substitute the known value found in step into one of the original equations to solve for the other variable. 7. Check the ordered pair in both equations. Eample Solving a Sstem b the Addition Method Solve the sstem b using the addition method. Multipl b. Multipl b. Step : Write both equations in standard form. There are no fractions or decimals. We ma choose to eliminate either variable. To eliminate, change the coefficients to and Step : Multipl the first equation b. Multipl the second equation b. Step : Add the equations. Step : Solve for. 0 The solution is (, ). Step 6: Substitute back into one of the original equations and solve for. Step 7: The ordered pair (, ) checks in both original equations.

21 Section. Solving Sstems of Equations b Using the Addition Method 97 Skill Practice. Solve b the addition method. TIP: To eliminate the variable in Eample, both equations were multiplied b appropriate constants to create and. We chose because it is the least common multiple of and. We could have solved the sstem b eliminating the -variable. To eliminate, we would multipl the top equation b and the bottom equation b. This would make the coefficients of the -variable 0 and 0, respectivel. Multipl b. Multipl b Eample Solving a Sstem of Equations b the Addition Method Solve the sstem b using the addition method Step : Write both equations in standard form Multipl b Step : Clear decimals. 6 0 Multipl b. 6 0 Step : Create opposite coefficients. Step : Add the equations. Step : Solve for Step 6: To solve for, substitute into one of the original equations. Skill Practice Answers. a, b

22 98 Chapter Sstems of Linear Equations Step 7: Check the ordered pair (0, ) in each original equation The solution is (0, ). Skill Practice. Solve b the addition method Solving Inconsistent Sstems and Dependent Sstems Eample Solving a Sstem of Equations b the Addition Method Solve the sstem b using the addition method Step : Equations are in standard form. 0 a b Step : Clear fractions. Multipl b Step : Multipl the first equation b. Step : Add the equations. Notice that both variables were eliminated. The sstem of equations is reduced to the identit 0 0. Therefore, the two original equations are equivalent and the sstem is dependent. The solution set consists of an infinite number of ordered pairs (, ) that fall on the common line of intersection 0 0, or equivalentl. The solution set can be written in set notation as, or e, ` f Skill Practice Answers. (0, )

23 0 Section. Solving Sstems of Equations b Using the Addition Method 99 Skill Practice. Solve b the addition method. Eample Solving an Inconsistent Sstem Solve the sstem b using the addition method. Standard form Step : Write the equations in standard form. Step : There are no decimals or fractions. Multipl b Step : Multipl the top equation b 0. Step : Add the equations. The equations reduce to a contradiction, indicating that the sstem has no solution. The sstem is inconsistent. The two equations represent parallel lines, as shown in Figure -8. There is no solution Figure -8 Skill Practice. Solve b the addition method Skill Practice Answers. Infinitel man solutions; {(, ) }; Dependent sstem. No solution; Inconsistent sstem

24 00 Chapter Sstems of Linear Equations Section. Boost our GRADE at mathzone.com! Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos Stud Skills Eercise. Instructors differ in what the emphasize on tests. For eample, test material ma come from the tetbook, notes, handouts, or homework. What does our instructor emphasize? Review Eercises For Eercises, use the slope-intercept form of the lines to determine the number of solutions for the sstem of equations Concept : The Addition Method For Eercises, solve the sstem b the addition method Concept : Solving Inconsistent Sstems and Dependent Sstems For Eercises, solve the sstems

25 Section. Solving Sstems of Equations b Using the Addition Method 0 Mied Eercises. Describe a situation in which ou would prefer to use the substitution method over the addition method.. If ou used the addition method to solve the given sstem, would it be easier to eliminate the - or - variable? Eplain. 7 0 For Eercises 0, solve b using either the addition method or the substitution method

26 0 Chapter Sstems of Linear Equations Epanding Your Skills For Eercises, use the addition method first to solve for. Then repeat the addition method again, using the original sstem of equations, this time solving for Section. Concepts. Applications Involving Cost. Applications Involving Mitures. Applications Involving Principal and Interest. Applications Involving Distance, Rate, and Time. Applications Involving Geometr Applications of Sstems of Linear Equations in Two Variables. Applications Involving Cost In Chapter we solved numerous application problems using equations that contained one variable. However, when an application has more than one unknown, sometimes it is more convenient to use multiple variables. In this section, we will solve applications containing two unknowns. When two variables are present, the goal is to set up a sstem of two independent equations. Eample Solving a Cost Application At an amusement park, five hot dogs and one drink cost $6. Two hot dogs and three drinks cost $9. Find the cost per hot dog and the cost per drink. Let h represent the cost per hot dog. Let d represent the cost per drink. Cost of of a b acost b $6 h d 6 hot dogs drink Cost of of a b acost b $9 h d 9 hot dogs drinks Label the variables. Write two equations. This sstem can be solved b either the substitution method or the addition method. We will solve b using the substitution method. The d-variable in the first equation is the easiest variable to isolate. h d 6 d h 6 h d 9 h h 6 9 h h 8 9 h 8 9 h 9 h Solve for d in the first equation. Substitute the quantit h 6 for d in the second equation. Clear parentheses. Solve for h. d 6 d Substitute h in the equation d h 6.

27 Section. Applications of Sstems of Linear Equations in Two Variables 0 Because h, the cost per hot dog is $.00. Because d, the cost per drink is $.00. A word problem can be checked b verifing that the solution meets the conditions specified in the problem. hot dogs drink ($.00) ($.00) $6.00 as epected hot dogs drinks ($.00) ($.00) $9.00 as epected Skill Practice. At the movie theater, Tom spent $7.7 on soft drinks and boes of popcorn. Carl bought soft drinks and bo of popcorn for total of $8.. Use a sstem of equations to find the cost of a soft drink and the cost of a bo of popcorn.. Applications Involving Mitures Eample Solving an Application Involving Chemistr One brand of cleaner used to etch concrete is % acid. A stronger industrialstrength cleaner is 0% acid. How man gallons of each cleaner should be mied to produce 0 gal of a 0% acid solution? Let represent the amount of % acid cleaner. Let represent the amount of 0% acid cleaner. % Acid 0% Acid 0% Acid Number of gallons of solution 0 Number of gallons of pure acid (0), or 8 From the first row of the table, we have Amount of amount of amount a b a b atotal % solution 0% solution of solution b 0 From the second row of the table we have Amount of amount of amount of pure acid in pure acid in pure acid in % solution 0% solution resulting solution Multipl b 00 to clear decimals Multipl b Create opposite coefficients of. Add the equations to eliminate. Skill Practice Answers. Soft drink: $.; popcorn: $.00

28 0 Chapter Sstems of Linear Equations Therefore, 8 gal of % acid solution must be added to gal of 0% acid solution to create 0 gal of a 0% acid solution. Skill Practice Substitute back into one of the original equations.. A pharmacist needs 8 ounces (oz) of a solution that is 0% saline. How man ounces of 60% saline solution and 0% saline solution must be mied to obtain the miture needed?. Applications Involving Principal and Interest Eample Solving a Miture Application Involving Finance Serena invested mone in two accounts: a savings account that ields.% simple interest and a certificate of deposit that ields 7% simple interest. The amount invested at 7% was twice the amount invested at.%. How much did Serena invest in each account if the total interest at the end of ear was $07.0? Let represent the amount invested in the savings account (the.% account). Let represent the amount invested in the certificate of deposit (the 7% account)..% Account 7% Account Total Principal Interest Because the amount invested at 7% was twice the amount invested at.%, we have Amount amount invested invested at 7% at.% From the second row of the table, we have Interest interest earned from earned from a total interest b % account 7% account Skill Practice Answers. 6 oz of 60% solution and oz of 0% solution 70,07,00 70,07,00 Multipl b 000 to clear decimals. Because the -variable in the first equation is isolated, we will use the substitution method. Substitute the quantit into the second equation.

29 Section. Applications of Sstems of Linear Equations in Two Variables 0 0,07,00 8,07,00,07, Solve for. Substitute 00 into the equation to solve for. Because 00, the amount invested in the savings account is $00. Because,000, the amount invested in the certificate of deposit is $,000. Check: $,000 is twice $00. Furthermore, Skill Practice 00,000 Interest interest earned from earned from $000.0 $, $07.0.% account 7% account. Seth invested mone in two accounts, one paing % interest and the other paing 6% interest. The amount invested at % was $000 more than the amount invested at 6%. He earned a total of $80 interest in ear. Use a sstem of equations to find the amount invested in each account.. Applications Involving Distance, Rate, and Time Eample Solving a Distance, Rate, and Time Application A plane flies 660 mi from Atlanta to Miami in. hr when traveling with a tailwind. The return flight against the same wind takes. hr. Find the speed of the plane in still air and the speed of the wind. Let p represent the speed of the plane in still air. Let w represent the speed of the wind. The speed of the plane with the wind: The speed of the plane against the wind: Set up a chart to organize the given information: (Plane s still airspeed) (wind speed): p w (Plane s still airspeed) (wind speed): p w Distance Rate Time With a tailwind 660 p w. Against a head wind 660 p w. Skill Practice Answers. $8000 invested at % and $7000 invested at 6%

30 06 Chapter Sstems of Linear Equations Two equations can be found b using the relationship d rt distance rate time. Distance speed time with with with 660 p w. wind wind wind Distance against wind speed against wind time against wind 660 p w. 660 p w. 660 p w. 660 p w. 660 p w. Notice that the first equation ma be divided b. and still leave integer coefficients. Similarl, the second equation ma be simplified b dividing b.. Divide b Divide b p w.. p w.. 0 p w 0 p w 0 p w 0 p w 990 p p 9 Add the equations. 0 9 w w Substitute p 9 into the equation 0 p w. Solve for w. The speed of the plane in still air is 9 mph, and the speed of the wind is mph. Skill Practice. A plane flies 00 mi from Orlando to New York in hr with a tailwind. The return flight against the same wind takes. hr. Find the speed of the plane in still air and the speed of the wind.. Applications Involving Geometr Eample Solving a Geometr Application Skill Practice Answers. Speed of plane: 0 mph; speed of wind: 60 mph The sum of the two acute angles in a right triangle is 90. The measure of one angle is 6 less than times the measure of the other angle. Find the measure of each angle. Let represent the measure of one acute angle. Let represent the measure of the other acute angle.

31 Section. Applications of Sstems of Linear Equations in Two Variables 07 The sum of the two acute angles is 90 : One angle is 6 less than times the other angle: Because one variable is alread isolated, we will use the substitution method Substitute 6 into the first equation. 6 To find, substitute into the equation The two acute angles in the triangle measure and 8. Skill Practice. Two angles are supplementar. The measure of one angle is 6 less than times the measure of the other. Use a sstem of equations to find the measures of the angles. Skill Practice Answers. 9 and Section. Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos. Make up a practice test for ourself. Use eamples or eercises from the tet. Be sure to cover each concept that was presented. Review Eercises. State three methods that can be used to solve a sstem of linear equations in two variables. For Eercises 6, state which method ou would prefer to use to solve the sstem. Then solve the sstem Concept : Applications Involving Cost 7. The local communit college theater put on a production of Chicago. There were 86 tickets sold, some for $6 (nonstudent price) and others for $ (student price). If the receipts for one performance totaled $60, how man of each tpe of ticket were sold? 8. John and Ariana bought school supplies. John spent $0.6 on notebooks and pens. Ariana spent $7.0 on notebooks and pens. What is the cost of notebook and what is the cost of pen?

32 08 Chapter Sstems of Linear Equations 9. Joe bought lunch for his fellow office workers on Monda. He spent $7. on hamburgers and fish sandwiches. Core bought lunch on Tuesda and spent $7. for hamburgers and fish sandwich. What is the price of hamburger, and what is the price of fish sandwich? 0. A group of four golfers pas $0 to pla a round of golf. Of these four, one is a member of the club and three are nonmembers. Another group of golfers consists of two members and one nonmember and pas a total of $7. What is the cost for a member to pla a round of golf, and what is the cost for a nonmember?. Meesha has a pocket full of change consisting of dimes and quarters. The total value is $.. There are 7 more quarters than dimes. How man of each coin are there?. Crstal has several dimes and quarters in her purse, totaling $.70. There is less dime than there are quarters. How man of each coin are there?. A coin collection consists of 0 pieces and $ coins. If there are coins worth $.0, how man 0 pieces and $ coins are there?. Suz has a pigg bank consisting of nickels and dimes. If there are 0 coins worth $.90, how man nickels and dimes are in the bank? Concept : Applications Involving Mitures. A jar of one face cream contains 8% moisturizer, and another tpe contains % moisturizer. How man ounces of each should be combined to get oz of a cream that is % moisturizer? 6. A chemistr student wants to mi an 8% acid solution with a % acid solution to get 6 L of a 6% acid solution. How man liters of the 8% solution and how man liters of the % solution should be mied? 7. How much pure bleach must be combined with a solution that is % bleach to make oz of a % bleach solution? 8. A fruit punch that contains % fruit juice is combined with a fruit drink that contains 0% fruit juice. How man ounces of each should be used to make 8 oz of a miture that is % fruit juice? Concept : Applications Involving Principal and Interest 9. Alina invested $7,000 in two accounts: one that pas % simple interest and one that pas % simple interest. At the end of the first ear, her total return was $68. How much was invested in each account? 0. Didi invested a total of $,000 into two accounts paing 7.% and 6% simple interest. If her total return at the end of the first ear was $80, how much did she invest in each account?. A credit union offers.% simple interest on a certificate of deposit (CD) and.% simple interest on a savings account. If Mr. Sorkin invested $00 more in the CD than in the savings account and the total interest after the first ear was $, how much was invested in each account?. Jod invested $000 less in an account paing % simple interest than she did in an account paing % simple interest. At the end of the first ear, the total interest from both accounts was $67. Find the amount invested in each account. Concept : Applications Involving Distance, Rate, and Time. It takes a boat hr to go 6 mi downstream with the current and hr to return against the current. Find the speed of the boat in still water and the speed of the current.

33 Section. Applications of Sstems of Linear Equations in Two Variables 09. The Gulf Stream is a warm ocean current that etends from the eastern side of the Gulf of Meico up through the Florida Straits and along the southeastern coast of the United States to Cape Hatteras, North Carolina. A boat travels with the current 00 mi from Miami, Florida, to Freeport, Bahamas, in. hr. The return trip against the same current takes hr. Find the speed of the boat in still water and the speed of the current.. A plane flew 70 mi in hr with the wind. It would take hr to travel the same distance against the wind. What is the rate of the plane in still air and the rate of wind? 6. Nikki and Tatiana rollerblade in opposite directions. Tatiana averages mph faster than Nikki. If the began at the same place and ended up 0 mi apart after hr, how fast did each of them travel? Concept : Applications Involving Geometr For Eercises 7, solve the applications involving geometr. If necessar, refer to the geometr formulas listed in the inside front cover of the tet. 7. In a right triangle, one acute angle measures 6 more than times the other. If the sum of the measures of the two acute angles must equal 90, find the measures of the acute angles. 8. An isosceles triangle has two angles of the same measure (see figure). If the angle represented b measures less than the angle, find the measures of all angles of the triangle. (Recall that the sum of the measures of the angles of a triangle is 80.) 9. Two angles are supplementar. One angle measures less than times the other. What are the measures of the two angles? 0. The measure of one angle is times the measure of another. If the two angles are supplementar, find the measures of the angles.. One angle measures more than twice another. If the two angles are complementar, find the measures of the angles.. Two angles are complementar. One angle measures more than times the measure of the other. What are the measures of the two angles? Mied Eercises. How much pure gold (K) must be mied with 60% gold to get 0 grams of 7% gold?. Two trains leave the depot at the same time, one traveling north and the other traveling south. The speed of one train is mph slower than the other. If after hr the distance between the trains is 90 miles, find the speed of each train.. There are two tpes of tickets sold at the Canadian Formula One Grand Pri race. The price of 6 grandstand tickets and general admissions tickets costs $0. The price of grandstand tickets and general admission tickets cost $00. What is the price of each tpe of ticket? 6. A granola mi contains % nuts. How man ounces of nuts must be added to get oz of granola with % nuts? 7. A bank offers two accounts, a mone market account at % simple interest and a regular savings account at.% interest. If Svetlana deposits $000 between the two accounts and receives $. total interest in the first ear, how much did she invest in each account? 8. A rectangle has the perimeter of m. The length is m longer than the width. Find the dimensions of the rectangle.

34 0 Chapter Sstems of Linear Equations 9. Kle rode his bike for one-half hour. He got a flat tire and had to walk for hr to get home. He rides his bike. mph faster than he walks. If the distance he traveled was 6. miles, what was his speed riding and what was his speed walking? 0. A basketball plaer scored 9 points b shooting two-point and three-point baskets. If she made a total of eight baskets, how man of each tpe did she make?. In a right triangle, the measure of one acute angle is one-fourth the measure of the other. Find the measures of the acute angles.. Angelo invested $8000 in two accounts: one that pas % and one that pas.8%. At the end of the first ear, his total interest earned was $. How much did he deposit in the account that pas %? Epanding Your Skills For Eercises 6, solve the business applications.. The demand for a certain printer cartridge is related to the price. In general, the higher the price, the lower the demand. The suppl for the printer cartridges is also related to price. The suppl and demand for the printer cartridges depend on the price according to the equations d 0 00 s 0 where is the price per cartridge in dollars and d is the demand measured in 000s of cartridges where is the price per cartridge in dollars and s is the suppl measured in 000s of cartridges Find the price at which the suppl and demand are in equilibrium (suppl demand), and confirm our answer with the graph.. The suppl and demand for a pack of note cards depend on the price according to the equations d s 90 where is the price per pack in dollars and where is the price per pack in dollars and note cards is the demand in 000s of note cards is the suppl measured in 000s of Find the price at which the suppl and demand are in equilibrium (suppl demand).. A rental car compan rents a compact car for $0 a da, plus $0. per mile. A midsize car rents for $0 a da, plus $0.0 per mile. a. Write a linear equation representing the cost to rent the compact car. b. Write a linear equation representing the cost to rent a midsize car. c. Find the number of miles at which the cost to rent either car would be the same. 6. One phone compan charges $0. per minute for long-distance calls. A second compan charges onl $0.0 per minute for long-distance calls, but adds a monthl fee of $.9. a. Write a linear equation representing the cost for the first compan. b. Write a linear equation representing the cost for the second compan. c. Find the number of minutes of long-distance calling for which the total bill from either compan would be the same. Suppl/Demand (000s) d s Suppl and Demand of Printer Cartridges Versus Price 00 d s Price per Cartridge ($)

35 Section. Sstems of Linear Equations in Three Variables and Applications Sstems of Linear Equations in Three Variables and Applications. Solutions to Sstems of Linear Equations in Three Variables In Sections.., we solved sstems of linear equations in two variables. In this section, we will epand the discussion to solving sstems involving three variables. A linear equation in three variables can be written in the form A B Cz D, where A, B, and C are not all zero. For eample, the equation z 6 is a linear equation in three variables. Solutions to this equation are ordered triples of the form (,, z) that satisf the equation. Some solutions to the equation z 6 are Check:,, 6, 0, 0 6 0,, 0 6 True True True Infinitel man ordered triples serve as solutions to the equation z 6. The set of all ordered triples that are solutions to a linear equation in three variables ma be represented graphicall b a plane in space. Figure -9 shows a portion of the plane z 6 in a -dimensional coordinate sstem. A solution to a sstem of linear equations in three variables is an ordered triple that satisfies each equation. Geometricall, a solution is a point of intersection of the planes represented b the equations in the sstem. A sstem of linear equations in three variables ma have one unique solution, infinitel man solutions, or no solution. z Figure -9 Section. Concepts. Solutions to Sstems of Linear Equations in Three Variables. Solving Sstems of Linear Equations in Three Variables. Applications of Linear Equations in Three Variables One unique solution (planes intersect at one point) The sstem is consistent. The sstem is independent. No solution (the three planes do not all intersect) The sstem is inconsistent. The sstem is independent.

36 Chapter Sstems of Linear Equations Infinitel man solutions (planes intersect at infinitel man points) The sstem is consistent. The sstem is dependent.. Solving Sstems of Linear Equations in Three Variables To solve a sstem involving three variables, the goal is to eliminate one variable. This reduces the sstem to two equations in two variables. One strateg for eliminating a variable is to pair up the original equations two at a time. Solving a Sstem of Three Linear Equations in Three Variables. Write each equation in standard form A B Cz D.. Choose a pair of equations, and eliminate one of the variables b using the addition method.. Choose a different pair of equations and eliminate the same variable.. Once steps and are complete, ou should have two equations in two variables. Solve this sstem b using the methods from Sections. and... Substitute the values of the variables found in step into an of the three original equations that contain the third variable. Solve for the third variable. 6. Check the ordered triple in each of the original equations. Eample Solving a Sstem of Linear Equations in Three Variables Solve the sstem. A z 7 B z C z z 7 z z Step : The equations are alread in standard form. It is often helpful to label the equations. The -variable can be easil eliminated from equations A and B and from equations A and C. This is accomplished b creating opposite coefficients for the -terms and then adding the equations.

37 Section. Sstems of Linear Equations in Three Variables and Applications Step : Eliminate the -variable from equations A and B. A z 7 B z Step : Eliminate the -variable again, this time from equations A and C. A z 7 C z Step : Now equations D and E can be paired up to form a linear sstem in two variables. Solve this sstem. D 7 z E z 8 Multipl b. Multipl b. Multipl b. Multipl b 7. 6z z 7 z D 6 9z z z 8 E 8 0z 8 77z 6 7z z Once one variable has been found, substitute this value into either equation in the two-variable sstem, that is, either equation D or E. D 7 z 7 Substitute z into equation D TIP: It is important to note that in steps and, the same variable is eliminated. A The solution is (,, ). Check: Skill Practice z Step : Step 6: Now that two variables are known, substitute these values for and z into an of the original three equations to find the remaining variable. Substitute and z into equation A. Check the ordered triple in the three original equations. z 7 7 True z z True True. Solve the sstem. z z z 8 Skill Practice Answers. (,, )

38 Chapter Sstems of Linear Equations Eample Appling Sstems of Linear Equations in Three Variables In a triangle, the smallest angle measures 0 more than one-half of the largest angle. The middle angle measures more than the smallest angle. Find the measure of each angle. Let represent the measure of the smallest z angle. Let represent the measure of the middle angle. Let z represent the measure of the largest angle. To solve for three variables, we need to establish three independent relationships among,, and z. A B C z 0 z 80 The smallest angle measures 0 more than onehalf the measure of the largest angle. The middle angle measures more than the measure of the smallest angle. The sum of the interior angles of a triangle measures 80. Clear fractions and write each equation in standard form. Standard Form A z 0 Multipl b. z 0 z 0 B C z 80 z 80 Notice equation B is missing the z-variable. Therefore, we can eliminate z again b pairing up equations A and C. A C z 0 z D B D Multipl b Pair up equations B and D to form a sstem of two variables. Solve for.

39 Section. Sstems of Linear Equations in Three Variables and Applications From equation From equation B C we have 7 9 we have z z 80 z 7 The smallest angle measures 7, the middle angle measures 9, and the largest angle measures 7. Skill Practice. The perimeter of a triangle is 0 in. The shortest side is in. shorter than the longest side. The longest side is 6 in. less than the sum of the other two sides. Find the length of each side. Eample Solving a Dependent Sstem of Linear Equations Solve the sstem. If there is not a unique solution, label the sstem as either dependent or inconsistent. A z 8 B z C z The first step is to make a decision regarding the variable to eliminate. The -variable is particularl eas to eliminate because the coefficients of in equations A and B are alread opposites. The -variable can be eliminated from equations B and C b multipling equation B b. A z 8 Pair up equations A and B to B z eliminate. z D B C Multipl b. z z 6 z z z E Pair up equations B and C to eliminate. Because equations D and E are equivalent equations, it appears that this is a dependent sstem. B eliminating variables we obtain the identit 0 0. D E z Multipl b. z z z 0 0 The result 0 0 indicates that there are infinitel man solutions and that the sstem is dependent. Skill Practice. Solve the sstem. If the sstem does not have a unique solution, identif the sstem as dependent or inconsistent. z 8 z 6 z 0 Skill Practice Answers. 8 in., 0 in., and in.. Dependent sstem

40 6 Chapter Sstems of Linear Equations Eample Solving an Inconsistent Sstem of Linear Equations Solve the sstem. If there is not a unique solution, identif the sstem as either dependent or inconsistent. 7z 6 z z 6 We will eliminate the -variable. A 7z Multipl b. 6 z 8 B 6 z 6 z C z (contradiction) The result 0 9 is a contradiction, indicating that the sstem has no solution. The sstem is inconsistent. Skill Practice. Solve the sstem. If the sstem does not have a unique solution, identif the sstem as dependent or inconsistent. z z 7 z 6. Applications of Linear Equations in Three Variables Eample Appling Sstems of Linear Equations to Nutrition Doctors have become increasingl concerned about the sodium intake in the U.S. diet. Recommendations b the American Medical Association indicate that most individuals should not eceed 00 mg of sodium per da. Liz ate slice of pizza, serving of ice cream, and glass of soda for a total of 00 mg of sodium. David ate slices of pizza, no ice cream, and glasses of soda for a total of 0 mg of sodium. Melinda ate slices of pizza, serving of ice cream, and glasses of soda for a total of 90 mg of sodium. How much sodium is in one serving of each item? Let represent the sodium content of slice of pizza. Let represent the sodium content of serving of ice cream. Let z represent the sodium content of glass of soda. From Liz s meal we have: A z 00 Skill Practice Answers. Inconsistent sstem From David s meal we have: From Melinda s meal we have: B C z 0

41 Section. Sstems of Linear Equations in Three Variables and Applications 7 Equation B is missing the -variable. Eliminating from equations A and C, we have A C Multipl b. Solve the sstem formed b equations B and D. B D z 00 z 90 z 0 z 880 From equation Multipl b. D D we have z 880 z 00 z 90 z 880 z 0 z z 880 z 0 From equation A we have z Therefore, slice of pizza has 660 mg of sodium, serving of ice cream has 0 mg of sodium, and glass of soda has 0 mg of sodium. Skill Practice. Annette, Barb, and Carlita work in a clothing shop. One da the three had combined sales of $80. Annette sold $0 more than Barb. Barb and Carlita combined sold $80 more than Annette. How much did each person sell? Skill Practice Answers. Annette sold $600, Barb sold $80, and Carlita sold $00. Section. Boost our GRADE at mathzone.com! Stud Skills Eercises. Look back over our notes for this chapter. Have ou highlighted the important topics? Have ou underlined the ke terms? Have ou indicated the places where ou are having trouble? If ou find that ou have problems with a particular topic, write a question that ou can ask our instructor either in class or in the office.. Define the ke terms. Practice Eercises Practice Problems Self-Tests NetTutor a. Linear equation in three variables b. Ordered triple e-professors Videos Review Eercises For Eercises, solve the sstems b using two methods: (a) the substitution method and (b) the addition method... 0

42 8 Chapter Sstems of Linear Equations. Two cars leave Kansas Cit at the same time. One travels east and one travels west. After hr the cars are 69 mi apart. If one car travels 7 mph slower than the other, find the speed of each car. Concept : Solutions to Sstems of Linear Equations in Three Variables 6. How man solutions are possible when solving a sstem of three equations with three variables? 7. Which of the following points are solutions to the sstem?,, 7,, 0, 6,, 0, z 0 z 0 z 8 9. Which of the following points are solutions to the sstem?,,,,,,,, z 6 z z 8. Which of the following points are solutions to the sstem?,,, 0, 0,,,, 6z 9 6 z 9 9z 0. Which of the following points are solutions to the sstem? 0,,,, 6, 0,,, z z z Concept : Solving Sstems of Linear Equations in Three Variables For Eercises, solve the sstem of equations.. z. z. z 7 z z z z 7z z 6. 6 z 7. z 8 6. z 0 z z z z 6 6z 9 6z 8 7. z 8. z 9. z z 7z 8 z 6 z 6 z z 0. z. z. z z 9 z z z z z 6 6z z 7z 6 Concept : Applications of Linear Equations in Three Variables. A triangle has one angle that measures more than twice the smallest angle, and the largest angle measures less than times the measure of the smallest angle. Find the measures of the three angles.

43 Section. Sstems of Linear Equations in Three Variables and Applications 9 6. The largest angle of a triangle measures less than times the measure of the smallest angle. The middle angle measures twice that of the smallest angle. Find the measures of the three angles. 7. The perimeter of a triangle is cm. The measure of the shortest side is 8 cm less than the middle side. The measure of the longest side is cm less than the sum of the other two sides. Find the lengths of the sides. 8. The perimeter of a triangle is ft. The longest side of the triangle measures 0 in. more than the shortest side. The middle side is times the measure of the shortest side. Find the lengths of the three sides in inches. 9. A movie theater charges $7 for adults, $ for children under age 7, and $ for seniors over age 60. For one showing of Batman the theater sold tickets and took in $8. If twice as man adult tickets were sold as the total of children and senior tickets, how man tickets of each kind were sold? 0. Goofie Golf has 8 holes that are par, par, or par. Most of the holes are par. In fact, there are times as man par s as par s. There are more par s than par s. How man of each tpe are there?. Combining peanuts, pecans, and cashews makes a part miture of nuts. If the amount of peanuts equals the amount of pecans and cashews combined, and if there are twice as man cashews as pecans, how man ounces of each nut is used to make 8 oz of part miture?. Souvenir hats, T-shirts, and jackets are sold at a rock concert. Three hats, two T-shirts, and one jacket cost $0. Two hats, two T-shirts, and two jackets cost $70. One hat, three T-shirts, and two jackets cost $80. Find the prices of the individual items.. In 00, Balor Universit in Waco, Teas, had twice as man students as Vanderbilt Universit in Nashville, Tennessee. Pace Universit in New York Cit had 800 more students than Vanderbilt Universit. If the enrollment for all three schools totaled 7,00, find the enrollment for each school.. Annie and Maria traveled overseas for seven das and staed in three different hotels in three different cities: Stockholm, Sweden; Oslo, Norwa; and Paris, France. The total bill for all seven nights (not including ta) was $00. The total ta was $06. The nightl cost (ecluding ta) to sta at the hotel in Paris was $80 more than the nightl cost (ecluding ta) to sta in Oslo. Find the cost per night for each hotel ecluding ta. Number Cost/Night Ta Cit of Nights ($) Rate Paris, 8% France Stockholm, % Sweden Oslo, z 0% Norwa Mied Eercises For Eercises, solve the sstem. If there is not a unique solution, label the sstem as either dependent or inconsistent.. z 6. z 7. z ( ) 6z z 7 ( ) 7 z 8. z z z 0 6 z z 6 z 8 z z z 8 z 9 8 z

44 0 Chapter Sstems of Linear Equations. z. z z z z z 0. 6z 0 7 z z z. Epanding Your Skills The sstems in Eercises 8 are called homogeneous sstems because each sstem has (0, 0, 0) as a solution. However, if a sstem is dependent, it will have infinitel man more solutions. For each sstem determine whether (0, 0, 0) is the onl solution or if the sstem is dependent.. 8z 0 6. z 0 7. z z 0 z 0 8 z 0 z 0 z 0 z 0 z 0 z 0 Section.6 Concepts. Introduction to Matrices. Solving Sstems of Linear Equations b Using the Gauss-Jordan Method Solving Sstems of Linear Equations b Using Matrices. Introduction to Matrices In Sections.,., and., we solved sstems of linear equations b using the substitution method and the addition method. We now present a third method called the Gauss-Jordan method that uses matrices to solve a linear sstem. A matri is a rectangular arra of numbers (the plural of matri is matrices). The rows of a matri are read horizontall, and the columns of a matri are read verticall. Ever number or entr within a matri is called an element of the matri. The order of a matri is determined b the number of rows and number of columns. A matri with m rows and n columns is an m n (read as m b n ) matri. Notice that with the order of a matri, the number of rows is given first, followed b the number of columns. Eample Determining the Order of a Matri Determine the order of each matri a. c b. c. 0 0 d. p 7 d a b c a. This matri has two rows and three columns. Therefore, it is a matri. b. This matri has four rows and one column. Therefore, it is a matri. A matri with one column is called a column matri.

45 Section.6 Solving Sstems of Linear Equations b Using Matrices c. This matri has three rows and three columns. Therefore, it is a matri. A matri with the same number of rows and columns is called a square matri. d. This matri has one row and three columns. Therefore, it is a matri. A matri with one row is called a row matri. Skill Practice Determine the order of the matri c 6 d A matri can be used to represent a sstem of linear equations written in standard form. To do so, we etract the coefficients of the variable terms and the constants within the equation. For eample, consider the sstem The matri A is called the coefficient matri. If we etract both the coefficients and the constants from the equations, we can construct the augmented matri of the sstem: A vertical bar is inserted into an augmented matri to designate the position of the equal signs. Eample Writing the Augmented Matri of a Sstem of Linear Equations Write the augmented matri for each linear sstem. a. b. z A c d c ` d z a. c ` d b Skill Practice Write the augmented matri for the sstem.. 6. z 8 TIP: Notice that zeros are inserted to denote the coefficient of each missing term. z 0 ` Skill Practice Answers..... c d

46 Chapter Sstems of Linear Equations c. Eample Writing a Linear Sstem from an Augmented Matri Write a sstem of linear equations represented b each augmented matri. a. c b. ` 8 6 d a. 8 b. 6 z z z c. 0 0z 0 0z or 0 0 z 0 z 0 Skill Practice Write a sstem of linear equations represented b each augmented matri. 7. c ` d Solving Sstems of Linear Equations b Using the Gauss-Jordan Method We know that interchanging two equations results in an equivalent sstem of linear equations. Interchanging two rows in an augmented matri results in an equivalent augmented matri. Similarl, because each row in an augmented matri represents a linear equation, we can perform the following elementar row operations that result in an equivalent augmented matri. Skill Practice Answers z 0 8 z 6 9.,, z 0 Elementar Row Operations The following elementar row operations performed on an augmented matri produce an equivalent augmented matri:. Interchange two rows.. Multipl ever element in a row b a nonzero real number.. Add a multiple of one row to another row. When we are solving a sstem of linear equations b an method, the goal is to write a series of simpler but equivalent sstems of equations until the solution is obvious. The Gauss-Jordan method uses a series of elementar row operations performed on the augmented matri to produce a simpler augmented matri. In

47 Section.6 Solving Sstems of Linear Equations b Using Matrices particular, we want to produce an augmented matri that has s along the diagonal of the matri of coefficients and 0s for the remaining entries in the matri of coefficients.a matri written in this wa is said to be written in reduced row echelon form. For eample, the augmented matri from Eample (c) is written in reduced row echelon form. The solution to the corresponding sstem of equations is easil recognized as,, and z 0. Similarl, matri B represents a solution of a and b. Eample B c 0 0 ` a b d Solving a Sstem of Linear Equations b Using the Gauss-Jordan Method Solve b using the Gauss-Jordan method. 0 c ` d R R c ` d R R R c 0 ` d R R c 0 ` d Set up the augmented matri. Switch row and row to get a in the upper left position. Multipl row b and add the result to row. This produces an entr of 0 below the upper left position. Multipl row b to produce a along the diagonal in the second row. R R R c 0 0 ` d The matri C is in reduced row echelon form. From the augmented matri, we have and. The solution to the sstem is (, ). C c 0 0 ` d Multipl row b and add the result to row. This produces a 0 in the first row, second column. Skill Practice 0. Solve b using the Gauss-Jordan method. Skill Practice Answers 0. (, 8)

48 Chapter Sstems of Linear Equations The order in which we manipulate the elements of an augmented matri to produce reduced row echelon form was demonstrated in Eample. In general, the order is as follows. First produce a in the first row, first column. Then use the first row to obtain 0s in the first column below this element. Net, if possible, produce a in the second row, second column. Use the second row to obtain 0s above and below this element. Net, if possible, produce a in the third row, third column. Use the third row to obtain 0s above and below this element. The process continues until reduced row echelon form is obtained. Eample Solving a Sstem of Linear Equations b Using the Gauss-Jordan Method Solve b using the Gauss-Jordan method. First write each equation in the sstem in standard form. z 0 z 0 6 7z 6 7z R R R R R R R R R R R R R R R R R R z 0 6 7z Set up the augmented matri. Multipl row b and add the result to row. Multipl row b and add the result to row. Multipl row b and add the result to row. Multipl row b and add the result to row. Multipl row b and add the result to row. Multipl row b and add the result to row. From the reduced row echelon form of the matri, we have,, and z. The solution to the sstem is (,, ).

49 Section.6 Solving Sstems of Linear Equations b Using Matrices Skill Practice Solve b using the Gauss-Jordan method.. z z z It is particularl eas to recognize a dependent or inconsistent sstem of equations from the reduced row echelon form of an augmented matri. This is demonstrated in Eamples 6 and 7. Eample 6 Solving a Dependent Sstem of Equations b Using the Gauss-Jordan Method Solve b using the Gauss-Jordan method. c ` d Set up the augmented matri. R R R c 0 0 ` 0 d Multipl row b result to row. and add the The second row of the augmented matri represents the equation 0 0; hence, the sstem is dependent. The solution is, 0 6. Skill Practice Solve b using the Gauss-Jordan method Eample 7 Solving an Inconsistent Sstem of Equations b Using the Gauss-Jordan Method Solve b using the Gauss-Jordan method. 9 R R R c 9 ` c 0 0 ` 7 d d Set up the augmented matri. Multipl row b and add the result to row. The second row of the augmented matri represents the contradiction 0 7; hence, the sstem is inconsistent. There is no solution. Skill Practice Answers. (,, ). Infinitel man solutions;, ; dependent sstem

50 6 Chapter Sstems of Linear Equations Skill Practice Answers. No solution; Inconsistent sstem Skill Practice. Solve b using the Gauss-Jordan method. 6 0 Calculator Connections Man graphing calculators have a matri editor in which the user defines the order of the matri and then enters the elements of the matri. For eample, the matri is entered as shown. D c ` 8 d Once an augmented matri has been entered into a graphing calculator, a rref function can be used to transform the matri into reduced row echelon form. Section.6 Boost our GRADE at mathzone.com! Stud Skills Eercises Practice Eercises Practice Problems Self-Tests NetTutor e-professors Videos. Prepare a one-page summar sheet with the most important information that ou need for the net test. On the da of the test, look at this sheet several times to refresh our memor, instead of tring to memorize new information.. Define the ke terms. a. Matri b. Order of a matri c. Column matri d. Square matri e. Row matri f. Coefficient matri g. Augmented matri h. Reduced row echelon form

51 Section.6 Solving Sstems of Linear Equations b Using Matrices 7 Review Eercises For Eercises, solve the sstem b using an method z 8. z z 7 z z 7 z Concept : Introduction to Matrices For Eercises 6, (a) determine the order of each matri and (b) determine if the matri is a row matri, a column matri, a square matri, or none of these c d 6. c c. 7 d. 9 8 d For Eercises 9, set up the augmented matri z 9. 6 z z 7 z 8 6z 6 8 z For Eercises 0, write a sstem of linear equations represented b the augmented matri c 6. c ` ` d 7 d Concept : Solving Sstems of Linear Equations b Using the Gauss-Jordan Method. Given the matri E E c 9 ` 8 7 d a. What is the element in the second row and third column? b. What is the element in the first row and second column?. Given the matri F 8 F c ` 0 d a. What is the element in the second row and second column? b. What is the element in the first row and third column?

52 8 Chapter Sstems of Linear Equations 6. Given the matri Z Z c ` d write the matri obtained b multipling the elements in the first row b. 8. Given the matri K K c ` d write the matri obtained b interchanging rows and. 0. Given the matri M M c write the matri obtained b multipling the first row b and adding the result to row.. Given the matri R ` 0 R 0 d 6 0 a. Write the matri obtained b multipling the first row b and adding the result to row. b. Using the matri obtained from part (a), write the matri obtained b multipling the first row b and adding the result to row. 7. Given the matri J J c 0 write the matri obtained b multipling the elements in the second row b. 9. Given the matri L 9 L c 7 ` d write the matri obtained b interchanging rows and.. Given the matri N N c ` 9 d write the matri obtained b multipling the first row b and adding the result to row.. Given the matri S 0 S 0 a. Write the matri obtained b multipling the first row b and adding the result to row. b. Using the matri obtained from part (a), write the matri obtained b multipling the first row b and adding the result to row. ` d For Eercises 9, solve the sstems b using the Gauss-Jordan method

53 Section.6 Solving Sstems of Linear Equations b Using Matrices 9 6. z 6 7. z 8. z 9. z 8z 6 z 0 z 0 z z 0z 6z z For Eercises 0, use the augmented matrices A, B, C, and D to answer true or false. A c 6 ` 7 d B c 6 ` C c ` D c 8 ` 7 d 0. The matri A is a matri.. Matri B is equivalent to matri A.. Matri A is equivalent to matri C.. Matri B is equivalent to matri D.. What does the notation R R mean when one is performing the Gauss-Jordan method?. What does the notation R R mean when one is performing the Gauss-Jordan method? 7 d 6. What does the notation R R R mean when one is performing the Gauss-Jordan method? 7. What does the notation R R R mean when one is performing the Gauss-Jordan method? 7 d Graphing Calculator Eercises For Eercises 8 6, use the matri features on a graphing calculator to epress each augmented matri in reduced row echelon form. Compare our results to the solution ou obtained in the indicated eercise. 8. c 9. c 60. ` ` 7 d d c 9 ` Compare with Eercise. Compare with Eercise. Compare with Eercise c ` d 0 Compare with Eercise 7. Compare with Eercise 6. Compare with Eercise 7. 6 d

54 0 Chapter Sstems of Linear Equations Section.7 Concepts. Introduction to Determinants. Determinant of a Matri. Cramer s Rule Determinants and Cramer s Rule. Introduction to Determinants Associated with ever square matri is a real number called the determinant of the matri. A determinant of a square matri A, denoted deta, is written b enclosing the elements of the matri within two vertical bars. For eample, If A c 6 0 d then det A ` 6 0 ` 0 0 If B 0 then det B Determinants have man applications in mathematics, including solving sstems of linear equations, finding the area of a triangle, determining whether three points are collinear, and finding an equation of a line between two points. The determinant of a matri is defined as follows: Determinant of a Matri The determinant of the matri c a b is the real number ad bc. It is written as c d d a b ` ad bc c d` Eample Evaluating a Determinant Evaluate the determinants. a. ` 6 ` b. ` 0 0 ` 6 a. ` For this determinant, a 6, b, c, and d `. ad bc 6a b 0 TIP: Eample (b) illustrates that the value of a determinant having a row of all zeros is 0. The same is true for a determinant having a column of all zeros. b. ` For this determinant, a, b, c 0, d ` ad bc

55 Section.7 Determinants and Cramer s Rule Skill Practice a. ` 8 b. ` ` Evaluate the determinants `. Determinant of a Matri To find the determinant of a matri, we first need to define the minor of an element of the matri. For an element of a matri, the minor of that element is the determinant of the matri obtained b deleting the row and column in which the element resides. For eample, consider the matri The minor of the element is found b deleting the first row and first column and then evaluating the determinant of the remaining matri: Now evaluate the determinant: ` ( 7)(6) ()() 6 ` For this matri, the minor for the element is. To find the minor of the element 7, delete the second row and second column, and then evaluate the determinant of the remaining matri Now evaluate the determinant: ` ()(6) (6)() 6 6 ` For this matri, the minor for the element 7 is 6. Eample Determining the Minor for Elements in a Matri Find the minor for each element in the first column of the matri. 0 6 For : The minor is: ` ( )( 6) ()() 9 6 ` 0 6 Skill Practice Answers a. 8 b. 0

56 Chapter Sstems of Linear Equations For : The minor is: ` ()( 6) ( )() 6 ` 0 6 For 0: The minor is: ` ()() ( )( ) 6 ` 0 6 Skill Practice. Find the minor for the element The determinant of a matri is defined as follows. Definition of a Determinant of a Matri a b c b a b c a ` c b ` a b a b c c ` c b ` a b c ` c ` b c From this definition, we see that the determinant of a matri can be written as a (minor of a ) a (minor of a ) a (minor of a ) Evaluating determinants in this wa is called epanding minors. Eample Evaluating a Determinant Evaluate the determinant ` ` ` ` ` 0 ` Skill Practice Answers 6. ` ` 6

57 Section.7 Determinants and Cramer s Rule Skill Practice. Evaluate the determinant. 9 6 Although we defined the determinant of a matri b epanding the minors of the elements in the first column, an row or column ma be used. However, we must choose the correct sign to appl to each term in the epansion. The following arra of signs is helpful. The signs alternate for each row and column, beginning with in the first row, first column. TIP: There is another method to determine the signs for each term of the epansion. For the a ij element, multipl the term b i j. Eample Evaluating a Determinant Evaluate the determinant, b epanding minors about the elements in the second row. 0 Signs obtained from the arra of signs 0 ` ` ` ` 0 ` ` TIP: Notice that the value of the determinant obtained in Eamples and is the same. Skill Practice. Evaluate the determinant In Eample, the third term in the epansion of minors was zero because the element 0 when multiplied b its minor is zero. To simplif the arithmetic in evaluating a determinant of a matri, epand about the row or column that has the most 0 elements. Skill Practice Answers..

58 Chapter Sstems of Linear Equations Calculator Connections The determinant of a matri can be evaluated on a graphing calculator. First use the matri editor to enter the elements of the matri. Then use a det function to evaluate the determinant. The determinant from Eamples and is evaluated below.. Cramer s Rule In Sections.,., and.6, we learned three methods to solve a sstem of linear equations: the substitution method, the addition method, and the Gauss-Jordan method. In this section, we will learn another method called Cramer s rule to solve a sstem of linear equations. Cramer s Rule for a Sstem of Linear Equations The solution to the sstem a b c a b c is given b D D and D D a where D ` b c `and D 0 D b ` b a ` D b ` c ` c a c a Eample Using Cramer s Rule to Solve a Sstem of Linear Equations Solve the sstem b using Cramer s rule. For this sstem: a b c a b c

59 Section.7 Determinants and Cramer s Rule D ` ` 9 D ` ` 8 D ` ` Therefore, D D 8 D D The solution is (, ). Check: Skill Practice. Solve using Cramer s rule. TIP: Here are some memor tips to help ou remember Cramer s rule. Coefficients of -terms -terms. The determinant D is the determinant of the coefficients of and. a D ` b ` a b -coefficients replaced b c and c. The determinant D has the column of -term coefficients replaced b c and c. c D ` b ` c b -coefficients replaced b c and c. The determinant D has the column of -term coefficients replaced b c and c. a D ` c ` c a It is important to note that the linear equations must be written in standard form to appl Cramer s rule. Eample 6 Using Cramer s Rule to Solve a Sstem of Linear Equations Solve the sstem b using Cramer s rule Skill Practice Answers. (, )

60 6 Chapter Sstems of Linear Equations Rewrite each equation in standard form. For this sstem: Therefore, The solution 8, checks in the original equations. Skill Practice a 0 b 6 c 7 a b 0 c D ` ` D ` ` D ` ` D D 6 8 D D Solve using Cramer s rule Cramer s rule can be used to solve a sstem of linear equations b using a similar pattern of determinants. Cramer s Rule for a Sstem of Linear Equations The solution to the sstem a b c z d a b c z d is given b a b c z d D D D D and z D z D a b c d b c where D a b c and D 0 D d b c a b c d b c Skill Practice Answers a d c a b d D a d c D z a b d a d c a b d 6. a, 6 b

61 Section.7 Determinants and Cramer s Rule 7 Eample 7 Using Cramer s Rule to Solve a Sstem of Linear Equations Solve the sstem b using Cramer s rule. z z z 6 D ` ` ` ` ` ` D ` 6 ` ` 6 ` ` ` 6 D z ` 6 ` ` 6 ` ` ` 6 Hence D D 7 7 D D 7 and z D z D 7 The solution is (,, ) D ` ` ` ` 6 ` ` Check: z z z 6 6 TIP: In Eample 7, we epanded the determinants about the first column. Skill Practice 7. Solve using Cramer s rule. z z z 6 Skill Practice Answers 7. (, 0, )

62 8 Chapter Sstems of Linear Equations Cramer s rule ma seem cumbersome for solving a sstem of linear equations. However, it provides convenient formulas that can be programmed into a computer or calculator to solve for,, and z. Cramer s rule can also be etended to solve a sstem of linear equations, a sstem of linear equations, and in general an n n sstem of linear equations. It is important to remember that Cramer s rule does not appl if D 0. In such a case, the sstem of equations is either dependent or inconsistent, and another method must be used to analze the sstem. Eample 8 Analzing a Dependent Sstem of Equations Solve the sstem. Use Cramer s rule if possible. Because D 0, Cramer s rule does not appl. Using the addition method to solve the sstem, we have Multipl b The solution is, Skill Practice D ` ` The sstem is dependent. Skill Practice Answers 8. No solution; Inconsistent sstem 8. Solve. Use Cramer s rule if possible. 6 Section.7 Boost our GRADE at mathzone.com! Stud Skills Eercise Practice Eercises Practice Problems Self-Tests NetTutor. Define the ke terms. a. Determinant b. Minor c. Cramer s rule e-professors Videos

63 Section.7 Determinants and Cramer s Rule 9 Concept : Introduction to Determinants For Eercises 7, evaluate the determinant of the matri.. `. ` 6. ` 8 `. ` 0 ` 6. ` 7. ` ` ` ` 8 ` Concept : Determinant of a Matri For Eercises 8, evaluate the minor corresponding to the given element from matri A. 8 A For Eercises, evaluate the minor corresponding to the given element from matri B Construct the sign arra for a matri. 7. Evaluate the determinant of matri B, using 8. Evaluate the determinant of matri C, using epansion b minors. epansion b minors. 0 B 6 0 B 9 a. About the first column a. About the first row C b. About the second row b. About the second column 9. When evaluating the determinant of a matri, eplain the advantage of being able to choose an row or column about which to epand minors. For Eercises 0, evaluate the determinants

64 0 Chapter Sstems of Linear Equations For Eercises 6, evaluate the determinants. 6. ` a 7. ` 8. ` b 8 ` 0 p 0 q r 0 s. z t 0 u a b 0 c f e 0 d c 0 b a 0 Concept : Cramer s Rule For Eercises, evaluate the determinants represented b D, D, and D For Eercises 0, solve the sstem b using Cramer s rule When does Cramer s rule not appl in solving a sstem of equations?. How can a sstem be solved if Cramer s rule does not appl? For Eercises 8, solve the sstem of equations b using Cramer s rule, if possible. If not possible, use another method For Eercises 9, solve for the indicated variable b using Cramer s rule. 9. z 9 0. z 8. z for z for z z 9 z 6 z z for z. z. 6z. 8 0 for 6 for 6z z 8 7 z 0 for z

65 Section.7 Determinants and Cramer s Rule For Eercises 8, solve the sstem b using Cramer s rule, if possible.. 6. z 7 z z 7. 8z 8. z z 0 8 z 6 z z 0 Epanding Your Skills For Eercises 9 6, solve the equation ` ` ` 8 7 ` 0 0 w 0 t 0 0 For Eercises 6 6, evaluate the determinant b using epansion b minors about the first column For Eercises 6 66, refer to the following sstem of four variables. z w 0 z w w 0 z 6. a. Evaluate the determinant D. 66. a. Evaluate the determinant D. b. Evaluate the determinant D. c. Solve for b computing. D D D b. Solve for b computing. D

66 Chapter Sstems of Linear Equations Chapter SUMMARY Section. Ke Concepts A sstem of linear equations in two variables can be solved b graphing. A solution to a sstem of linear equations is an ordered pair that satisfies each equation in the sstem. Graphicall, this represents a point of intersection of the lines. There ma be one solution, infinitel man solutions, or no solution. Solving Sstems of Equations b Graphing Eamples Eample Solve b graphing. Write each equation in m b form to graph. 0 One solution Infinitel man No solution Consistent solutions Inconsistent Independent Consistent Independent Dependent A sstem of equations is consistent if there is at least one solution. A sstem is inconsistent if there is no solution. A linear sstem in and is dependent if two equations represent the same line. The solution set is the set of all points on the line. If two linear equations represent different lines, then the sstem of equations is independent. The solution is the point of intersection (, ). 0 (, )

67 v Summar Section. Ke Concepts Substitution Method. Isolate one of the variables.. Substitute the quantit found in step into the other equation.. Solve the resulting equation.. Substitute the value from step back into the equation from step to solve for the remaining variable.. Check the ordered pair in both equations, and write the answer as an ordered pair. Solving Sstems of Equations b Using the Substitution Method Eamples Eample Isolate a variable. Substitute Now solve for. The solution is (, ) and checks in both equations.

68 Chapter Sstems of Linear Equations Section. Ke Concepts Addition Method. Write both equations in standard form A B C.. Clear fractions or decimals (optional).. Multipl one or both equations b nonzero constants to create opposite coefficients for one of the variables.. Add the equations from step to eliminate one variable.. Solve for the remaining variable. 6. Substitute the known value from step back into one of the original equations to solve for the other variable. 7. Check the ordered pair in both equations. Solving Sstems of Equations b Using the Addition Method Eamples Eample Mult. b. Mult. b The solution is (, ) and checks in both equations. A sstem is consistent if there is at least one solution. A sstem is inconsistent if there is no solution. An inconsistent sstem is detected b a contradiction (such as 0. A sstem is independent if the two equations represent different lines. A sstem is dependent if the two equations represent the same line. This produces infinitel man solutions. A dependent sstem is detected b an identit (such as 0 0. Eample 0 7 Contradiction. There is no solution. The sstem is inconsistent. Eample 6 Mult. b. Mult. b Identit. There are infinitel man solutions. The sstem is dependent.

69 Summar Section. Ke Concepts Solve application problems b using sstems of linear equations in two variables. Cost applications Miture applications Applications involving principal and interest Applications involving distance, rate, and time Geometr applications Steps to Solve Applications:. Label two variables.. Construct two equations in words.. Write two equations.. Solve the sstem.. Write the answer in words. Applications of Sstems of Linear Equations in Two Variables Eamples Eample Mercedes invested $00 more in a certificate of deposit that pas 6.% simple interest than she did in a savings account that pas % simple interest. If her total interest at the end of ear is $6.0, find the amount she invested in the 6.% account. Let represent the amount of mone invested at 6.%. Let represent the amount of mone invested at %. a Amount invested amount invested b a b $00 at 6.% at % Interest earned interest earned from 6.% from % $6.0 account account Using substitution gives Mercedes invested $600 at 6.% and $000 at %.

70 6 Chapter Sstems of Linear Equations Section. Ke Concepts A linear equation in three variables can be written in the form A B Cz D, where A, B, and C are not all zero. The graph of a linear equation in three variables is a plane in space. Sstems of Linear Equations in Three Variables and Applications Eamples Eample A B C z z z 7 A solution to a sstem of linear equations in three variables is an ordered triple that satisfies each equation. Graphicall, a solution is a point of intersection among three planes. A sstem of linear equations in three variables ma have one unique solution, infinitel man solutions (dependent sstem), or no solution (inconsistent sstem). A D E and B z z 9 D A and C z 8 z 7 7 E Substitute into either equation D or E. D 9 8 Substitute and into equation A, B, or C. A z z 0 The solution is (,, 0).

71 Summar 7 Section.6 Ke Concepts A matri is a rectangular arra of numbers displaed in rows and columns. Ever number or entr within a matri is called an element of the matri. The order of a matri is determined b the number of rows and number of columns. A matri with m rows and n columns is an m n matri. Solving Sstems of Linear Equations b Using Matrices Eamples Eample [ ] is a matri (called a row matri). c 8 d is a matri (called a square matri). c is a matri (called a column matri). d A sstem of equations written in standard form can be represented b an augmented matri consisting of the coefficients of the terms of each equation in the sstem. Eample The augmented matri for is c 6 ` 6 d The Gauss-Jordan method can be used to solve a sstem of equations b using the following elementar row operations on an augmented matri.. Interchange two rows.. Multipl ever element in a row b a nonzero real number.. Add a multiple of one row to another row. These operations are used to write the matri in reduced row echelon form. c 0 0 ` a b d which represents the solution, a and b. Eample Solve the sstem from Eample b using the Gauss-Jordan method. R c 6 R ` d R c 6 R R ` d 9R R R R R c 0 c 0 0 ` ` 6 d d and

72 8 Chapter Sstems of Linear Equations Section.7 Ke Concepts The determinant of matri A a b is denoted deta `. c d ` The determinant of a matri is defined as a ` c b d ` = ad bc. The determinant of a matri is defined b c a c b d d a b c b a b c a ` c b ` a ` c ` b a b c c b c Determinants and Cramer s Rule b a ` c ` c b Eamples Eample For A deta Eample For detb c 7 d, 7 ` ` B , ` ` 0 ` ` 6 ` ` Cramer s rule can be used to solve a sstem of linear equations. a b c is given b D D a b c and a where D ` b ` (and D 0), b c D ` b ` b c a D D a and D ` c `. c a Eample Solve D ` `, D 0 8 ` ` 8, 0 D ` ` Therefore, 8,

73 Review Eercises 9 Chapter Review Eercises Section.. Determine if the ordered pair is a solution to the sstem. 7 a. (, ) b., For Eercises, answer true or false.. An inconsistent sstem has one solution. Section. For Eercises 8, solve the sstems b using the substitution method Parallel lines form an inconsistent sstem.. Lines with different slopes intersect in one point. For Eercises 7, solve the sstem b graphing Section. For Eercises, solve the sstems b using the addition method

74 0 Chapter Sstems of Linear Equations Section.. Antonio invested twice as much mone in an account paing % simple interest as he did in an account paing.% simple interest. If his total interest at the end of ear is $0.7, find the amount he invested in the % account.. A school carnival sold tickets to ride on a Ferris wheel. The charge was $.0 for adults and $.00 for students. If tickets were sold for a total of $70.0, how man of each tpe of ticket were sold?. How man liters of 0% saline solution must be mied with 0% saline solution to produce 6 L of a.% saline solution?. It takes a pilot hr to travel with the wind to get from Jacksonville, Florida, to Mrtle Beach, South Carolina. Her return trip takes hr fling against the wind. What is the speed of the wind and the speed of the plane in still air if the distance between Jacksonville and Mrtle Beach is 80 mi? 6. Two phone companies offer discount rates to students. Compan : $9.9 per month, plus $0.0 per minute for long-distance calls Compan : $.9 per month, plus $0.08 per minute for long-distance calls a. Write a linear equation describing the total cost,, for min of long-distance calls from Compan. b. Write a linear equation describing the total cost,, for min of long-distance calls from Compan. c. How man minutes of long-distance calls would result in equal cost for both offers? 7. Two angles are complementar. One angle measures 6 more than times the measure of the other. What are the measures of the two angles? Section. For Eercises 8, solve the sstems of equations. If a sstem does not have a unique solution, label the sstem as either dependent or inconsistent. 8. z 0 9. z 0 6 z z 6 6z. The perimeter of a right triangle is 0 ft. One leg is ft longer than twice the shortest leg. The hpotenuse is ft less than times the shortest leg. Find the lengths of the sides of this triangle.. Three pumps are working to drain a construction site. Working together, the pumps can pump 90 gal/hr of water. The slowest pump pumps 0 gal/hr less than the fastest pump. The fastest pump pumps 0 gal/hr less than the sum of the other two pumps. How man gallons can each pump drain per hour? Section.6 For Eercises 7, determine the order of each matri For Eercises 8 9, set up the augmented matri z z z 6 z 8 0. z. z z 8 z z 9

75 Review Eercises For Eercises 0, write a corresponding sstem of equations from the augmented matri. 0. c 0 9 `. 0 d. Given the matri C a. What is the element in the second row and first column? b. Write the matri obtained b multipling the first row b and adding the result to row.. Given the matri D a. Write the matri obtained b multipling the first row b and adding the result to row. b. Using the matri obtained in part (a), write the matri obtained b multipling the first row b and adding the result to row. For Eercises 7, solve the sstem b using the Gauss-Jordan method z 7. z 9 z C c ` D 6 d z z 8 8 z 9 For Eercises, evaluate the minor corresponding to the given element from matri A For Eercises 6 9, evaluate the determinant For Eercises 60 6, solve the sstem using Cramer s rule z 7 6. z 0 For Eercises 66 67, solve the sstem of equations using Cramer s rule if possible. If not possible, use another method A 6 z z 0 6 z z z 6 Section.7 For Eercises 8, evaluate the determinant. 8. ` 9. ` 6 ` 0 0 ` 9 0. `. ` ` 8 `

76 Chapter Sstems of Linear Equations Chapter Test. Determine if the ordered pair, is a solution to the sstem. 7 Match each figure with the appropriate description Solve the sstem b using the substitution method Solve the sstem b using the addition method. 6 8 For Eercises 8, solve the sstem of equations a. The sstem is consistent and dependent. There are infinitel man solutions. b. The sstem is consistent and independent. There is one solution. c. The sstem is inconsistent and independent. There are no solutions.. Solve the sstem b graphing. 7.. z 6 z 9 z z 6 z z 8. How man liters of a 0% acid solution should be mied with a 60% acid solution to produce 00 L of a % acid solution? 6. Two angles are complementar. Two times the measure of one angle is 60 less than the measure of the other. Find the measure of each angle. 7. Working together, Joanne, Kent, and Geoff can process 0 orders per da for their business. Kent can process 0 more orders per da than Joanne can process. Geoff can process 0 fewer orders per da than Kent and Joanne combined. Find the number of orders that each person can process per da.

77 Cumulative Review Eercises 8. Write an eample of a matri. For Eercises, find the determinant of the matri. 9. Given the matri A a. Write the matri obtained b multipling the first row b and adding the result to row. b. Using the matri obtained in part (a), write the matri obtained b multipling the first row b and adding the result to row. For Eercises 0, solve b using the Gauss-Jordan method A z 0 z.. 0 c d 0 0 For Eercises, use Cramer s rule for solve for.. 6. z 9 6. Solve the sstem: z Chapters Cumulative Review Eercises For Eercises, solve the equation. For Eercises 6 7, graph the lines a a 6 6a b. Simplif the epression. a b 0. Solve the inequalit. Write the answer in interval notation. 6. Identif the slope and the - and -intercepts of the line. 8. Find the slope of the line passing through the points, 0 and 6, Find an equation for the line that passes through the points, 8 and,. Write the answer in slope-intercept form.

78 Chapter Sstems of Linear Equations 0. Solve the sstem b using the addition method. 6. Solve the sstem b using the substitution method.. A child s pigg bank contains 9 coins consisting of nickels, dimes, and quarters. The total amount of mone in the bank is $.0. If the number of quarters is more than twice the number of nickels, find the number of each tpe of coin in the bank.. Two video clubs rent tapes according to the following fee schedules: Club : $ initiation fee plus $.0 per tape Club : $0 initiation fee plus $.00 per tape a. Write a linear equation describing the total cost,, of renting tapes from club. b. Write a linear equation describing the total cost,, of renting tapes from club. c. How man tapes would have to be rented to make the cost for club the same as the cost for club?. Solve the sstem. z 7z z 6. Determine the order of the matri. c 6 0 d 6. Write an eample of a matri. 7. List at least two different row operations. 8. Solve the sstem b using the Gauss-Jordan method. 9. Find the determinant of matri C c 8 d. 0. Solve the sstem using Cramer s rule.