3330_070.qd 96 /5/05 Chapter 7 7. 9:39 AM Page 96 Sstems of Equations and Inequalities Linear and Nonlinear Sstems of Equations What ou should learn Use the method of substitution to solve sstems of linear equations in two variables. Use the method of substitution to solve sstems of nonlinear equations in two variables. Use a graphical approach to solve sstems of equations in two variables. Use sstems of equations to model and solve real-life problems. Wh ou should learn it Graphs of sstems of equations help ou solve real-life problems. For instance, in Eercise 7 on page 505, ou can use the graph of a sstem of equations to approimate when the consumption of wind energ eceeded the consumption of solar energ. The Method of Substitution Up to this point in the tet, most problems have involved either a function of one variable or a single equation in two variables. However, man problems in science, business, and engineering involve two or more equations in two or more variables. To solve such problems, ou need to find solutions of a sstem of equations. Here is an eample of a sstem of two equations in two unknowns. 3 5 Equation A solution of this sstem is an ordered pair that satisfies each equation in the sstem. Finding the set of all solutions is called solving the sstem of equations. For instance, the ordered pair, is a solution of this sstem. To check this, ou can substitute for and for in each equation. Check (, ) in and Equation : 5? 5 5 3? 3 6 Write. Substitute for and for. checks in. Write Equation. Substitute for and for. checks in Equation. In this chapter, ou will stud four was to solve sstems of equations, beginning with the method of substitution... 3.. ML Sinibaldi /Corbis Method Substitution Graphical method Elimination Gaussian elimination Section 7. 7. 7. 7.3 Tpe of Sstem Linear or nonlinear, two variables Linear or nonlinear, two variables Linear, two variables Linear, three or more variables Method of Substitution. Solve one of the equations for one variable in terms of the other.. Substitute the epression found in Step into the other equation to obtain an equation in one variable. 3. Solve the equation obtained in Step. The HM mathspace CD-ROM and Eduspace for this tet contain additional resources related to the concepts discussed in this chapter.. Back-substitute the value obtained in Step 3 into the epression obtained in Step to find the value of the other variable. 5. Check that the solution satisfies each of the original equations.
3330_070.qd /5/05 9:39 AM Page 97 Eploration Eample Use a graphing utilit to graph and in the same viewing window. Use the zoom and trace features to find the coordinates of the point of intersection. What is the relationship between the point of intersection and the solution found in Eample? Because man steps are required to solve a sstem of equations, it is ver eas to make errors in arithmetic. So, ou should alwas check our solution b substituting it into each equation in the original sstem. Solving a Sstem of Equations b Substitution Solve the sstem of equations. Equation Begin b solving for in. Solve for in. Net, substitute this epression for into Equation and solve the resulting single-variable equation for. 6 3 Write Equation. Substitute for. Distributive Propert Combine like terms. Divide each side b. Finall, ou can solve for b back-substituting 3 into the equation, to obtain Write revised. Substitute 3 for. Solve for. The solution is the ordered pair 3,. You can check this solution as follows. Check 3. Section 7. Linear and Nonlinear Sstems of Equations 97 Substitute 3, into : Write. 3? Substitute for and. checks in. Substitute 3, into Equation : Write Equation. 3? Substitute for and. checks in Equation. Because 3, satisfies both equations in the sstem, it is a solution of the sstem of equations. Now tr Eercise 5. The term back-substitution implies that ou work backwards. First ou solve for one of the variables, and then ou substitute that value back into one of the equations in the sstem to find the value of the other variable.
3330_070.qd /5/05 9:39 AM Page 98 98 Chapter 7 Sstems of Equations and Inequalities You ma want to compare and contrast solving a problem using a sstem of two equations with solving a problem using one equation. Eample Solving a Sstem b Substitution A total of $,000 is invested in two funds paing 5% and 3% simple interest. (Recall that the formula for simple interest is I Prt, where P is the principal, r is the annual interest rate, and t is the time.) The earl interest is $500. How much is invested at each rate? Verbal Model: 5% fund 5% interest 3% fund 3% interest Total investment Total interest When using the method of substitution, it does not matter which variable ou choose to solve for first. Whether ou solve for first or first, ou will obtain the same solution. When making our choice, ou should choose the variable and equation that are easier to work with. For instance, in Eample, solving for in is easier than solving for in Equation. One wa to check the answers ou obtain in this section is to use a graphing utilit. For instance, enter the two equations in Eample,000 Technolog 500 0.05 0.03 and find an appropriate viewing window that shows where the two lines intersect. Then use the intersect feature or the zoom and trace features to find the point of intersection. Does this point agree with the solution obtained at the right? Labels: Amount in 5% fund (dollars) Interest for 5% fund 0.05 (dollars) Amount in 3% fund (dollars) Interest for 3% fund 0.03 (dollars) Total investment,000 (dollars) Total interest 500 (dollars) Sstem:,000 0.05 0.03 500 Equation To begin, it is convenient to multipl each side of Equation b 00. This eliminates the need to work with decimals. 000.05 0.03 00500 5 3 50,000 Multipl each side b 00. Revised Equation To solve this sstem, ou can solve for in.,000 Then, substitute this epression for resulting equation for. 5 3 50,000 5,000 3 50,000 Revised into revised Equation and solve the Write revised Equation. Substitute,000 for. 60,000 5 3 50,000 Distributive Propert 0,000 Combine like terms. 5000 Divide each side b. Net, back-substitute the value 5000 to solve for.,000 Write revised.,000 5000 Substitute 5000 for. 7000 Simplif. The solution is 7000, 5000. So, $7000 is invested at 5% and $5000 is invested at 3%. Check this in the original sstem. Now tr Eercise 9.
3330_070.qd /5/05 9:39 AM Page 99 Section 7. Linear and Nonlinear Sstems of Equations 99 Nonlinear Sstems of Equations The equations in Eamples and are linear. The method of substitution can also be used to solve sstems in which one or both of the equations are nonlinear. Eample 3 Substitution: Two- Case Eploration Use a graphing utilit to graph the two equations in Eample 3 7 in the same viewing window. How man solutions do ou think this sstem has? Repeat this eperiment for the equations in Eample. How man solutions does this sstem have? Eplain our reasoning. Point out that it is not practical to solve Equation for (instead of for ) in Eample 3. Solve the sstem of equations. 7 Equation Begin b solving for in Equation to obtain. Net, substitute this epression for into and solve for. 7 7 8 0 0, Substitute for into. Simplif. Write in general form. Factor. Solve for. Back-substituting these values of to solve for the corresponding values of produces the solutions, 7 and, 5. Check these in the original sstem. Now tr Eercise 5. When using the method of substitution, ou ma encounter an equation that has no solution, as shown in Eample. Eample Substitution: No-Real- Case Solve the sstem of equations. 3 Equation Begin b solving for in to obtain. Net, substitute this epression for into Equation and solve for. 3 0 ± 3 Substitute for into Equation. Simplif. Use the Quadratic Formula. Because the discriminant is negative, the equation 0 has no (real) solution. So, the original sstem has no (real) solution. Now tr Eercise 7.
3330_070.qd /5/05 9:39 AM Page 500 500 Chapter 7 Sstems of Equations and Inequalities Technolog Most graphing utilities have builtin features that approimate the point(s) of intersection of two graphs. Tpicall, ou must enter the equations of the graphs and visuall locate a point of intersection before using the intersect feature. Use this feature to find the points of intersection of the graphs in Figures 7. to 7.3. Be sure to adjust our viewing window so that ou see all the points of intersection. Graphical Approach to Finding s From Eamples, 3, and, ou can see that a sstem of two equations in two unknowns can have eactl one solution, more than one solution, or no solution. B using a graphical method, ou can gain insight about the number of solutions and the location(s) of the solution(s) of a sstem of equations b graphing each of the equations in the same coordinate plane. The solutions of the sstem correspond to the points of intersection of the graphs. For instance, the two equations in Figure 7. graph as two lines with a single point of intersection; the two equations in Figure 7. graph as a parabola and a line with two points of intersection; and the two equations in Figure 7.3 graph as a line and a parabola that have no points of intersection. (, 0) + 3 = = 3 = (, ) = 3 (0, ) + = 3 3 + = 3 One intersection point Two intersection points No intersection points FIGURE 7. FIGURE 7. FIGURE 7.3 Eample 5 Solving a Sstem of Equations Graphicall Solve the sstem of equations. ln Equation + = = ln Sketch the graphs of the two equations. From the graphs of these equations, it is clear that there is onl one point of intersection and that, 0 is the solution point (see Figure 7.). You can confirm this b substituting for and 0 for in both equations. (, 0) Check (, 0) in : ln Write. 0 ln checks. FIGURE 7. Check (, 0) in Equation : Write Equation. 0 Equation checks. Now tr Eercise 33. Eample 5 shows the value of a graphical approach to solving sstems of equations in two variables. Notice what would happen if ou tried onl the substitution method in Eample 5. You would obtain the equation ln. It would be difficult to solve this equation for using standard algebraic techniques.
3330_070.qd /5/05 9:39 AM Page 50 Section 7. Linear and Nonlinear Sstems of Equations 50 Activities:. Solve the sstem b the method of substitution. 3 6 0 Answer: (, ). Find all points of intersection. 5 0 8 5 0 Answer: 0, 5,, 3. Solve the sstem graphicall. 3 6 ln Answer: (, 0) Applications The total cost C of producing units of a product tpicall has two components the initial cost and the cost per unit. When enough units have been sold so that the total revenue R equals the total cost C, the sales are said to have reached the break-even point. You will find that the break-even point corresponds to the point of intersection of the cost and revenue curves. Eample 6 Break-Even Analsis A shoe compan invests $300,000 in equipment to produce a new line of athletic footwear. Each pair of shoes costs $5 to produce and is sold for $60. How man pairs of shoes must be sold before the business breaks even? The total cost of producing units is Total cost Cost per unit C 5 300,000. Number of units The revenue obtained b selling units is Initial cost Total revenue Price per unit Number of units Revenue and cost (in dollars) 600,000 500,000 00,000 300,000 00,000 00,000 FIGURE 7.5 Break-Even Analsis Break-even point: 555 units R = 60 Loss Number of units Profit C = 5 + 300,000 3,000 6,000 9,000 R 60. Equation Because the break-even point occurs when R C, ou have C 60, and the sstem of equations to solve is C 5 300,000. C 60 Now ou can solve b substitution. 60 5 300,000 Substitute 60 for C in. 55 300,000 Subtract 5 from each side. 555 Divide each side b 55. So, the compan must sell about 555 pairs of shoes to break even. Note in Figure 7.5 that revenue less than the break-even point corresponds to an overall loss, whereas revenue greater than the break-even point corresponds to a profit. Now tr Eercise 63. Another wa to view the solution in Eample 6 is to consider the profit function P R C. The break-even point occurs when the profit is 0, which is the same as saing that R C.
3330_070.qd /5/05 9:39 AM Page 50 50 Chapter 7 Sstems of Equations and Inequalities Eample 7 Movie Ticket Sales The weekl ticket sales for a new comed movie decreased each week. At the same time, the weekl ticket sales for a new drama movie increased each week. Models that approimate the weekl ticket sales S (in millions of dollars) for each movie are S 60 8 Comed S 0.5 Drama where represents the number of weeks each movie was in theaters, with 0 corresponding to the ticket sales during the opening weekend. After how man weeks will the ticket sales for the two movies be equal? Algebraic Because the second equation has alread been solved for S in terms of, substitute this value into the first equation and solve for, as follows. 0.5 60 8 Substitute for S in..5 8 60 0 Add 8 and 0 to each side..5 50 Combine like terms. Divide each side b.5. So, the weekl ticket sales for the two movies will be equal after weeks. Now tr Eercise 65. Numerical You can create a table of values for each model to determine when the ticket sales for the two movies will be equal. Number of weeks, Sales, S (comed) Sales, S (drama) 0 3 5 6 60 5 36 8 0 0.5 9 3.5 8 3.5 37 So, from the table above, ou can see that the weekl ticket sales for the two movies will be equal after weeks. Writing About Mathematics Suggestion: Consider asking our students to make up their own problems that involve a choice of solution methods. Students can then trade their problems with other members of the class. Make sure that students are able to interpret attributes such as slopes and points of intersection. W RITING ABOUT MATHEMATICS Interpreting Points of Intersection You plan to rent a -foot truck for a two-da local move. At truck rental agenc A, ou can rent a truck for $9.95 per da plus $0.9 per mile. At agenc B, ou can rent a truck for $50 per da plus $0.5 per mile. a. Write a total cost equation in terms of and for the total cost of renting the truck from each agenc. b. Use a graphing utilit to graph the two equations in the same viewing window and find the point of intersection. Interpret the meaning of the point of intersection in the contet of the problem. c. Which agenc should ou choose if ou plan to travel a total of 00 miles during the two-da move? Wh? d. How does the situation change if ou plan to drive 00 miles during the two-da move?
3330_070.qd /5/05 9:39 AM Page 503 Section 7. Linear and Nonlinear Sstems of Equations 503 7. Eercises The HM mathspace CD-ROM and Eduspace for this tet contain step-b-step solutions to all odd-numbered eercises. The also provide Tutorial Eercises for additional help. Eercises containing sstems with no solutions: 3,, 7, 39, 0, 50, 53, 5 VOCABULARY CHECK: Fill in the blanks.. A set of two or more equations in two or more variables is called a of.. A of a sstem of equations is an ordered pair that satisfies each equation in the sstem. 3. Finding the set of all solutions to a sstem of equations is called the sstem of equations.. The first step in solving a sstem of equations b the method of is to solve one of the equations for one variable in terms of the other variable. 5. Graphicall, the solution of a sstem of two equations is the of of the graphs of the two equations. 6. In business applications, the point at which the revenue equals costs is called the point. PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.eduspace.com. In Eercises, determine whether each ordered pair is a solution of the sstem of equations... 3.. 6 6 3 In Eercises 5, solve the sstem b the method of substitution. Check our solution graphicall. 6 e 3 log 3 8 9 9 6 7. 8. 6 (a) 0, 3 (c) 3, (a), 3 (c) 3, 3 3 (a), 0 (c) 0, 3 (a) 9, 37 9 (c), 3 5. 6 6. 0 5 3 3 0 8 6 (b), (d), 3 (b), 9 (d) 7, 37 (b) 0, (d), (b) 0, (d), 6 9. 5 0. 5 6 6 8. 0. 0 3.. 3 3 3 8 6 6 0 3 5 0 3 3 3
3330_070.qd /5/05 9:39 AM Page 50 50 Chapter 7 Sstems of Equations and Inequalities In Eercises 5 8, solve the sstem b the method of substitution. 5. 0 6. 5 3 0 5 3 7. 0 8. 6 3 0 5 0 0 9..5 0.8.3 0. 0.5 3. 9.0 0.3 0. 0. 0..6 3.6.. 3 0 5 8 3 0 3.. 3 6 5 3 5 6 7 3 6 5. 6. 0 0 0 3 0 7. 8. 3 3 In Eercises 9, solve the sstem graphicall. 9. 30. 0 3 5 3 0 3. 3 3. 5 3 7 33. 3. 35. 3 0 36. 7 37. 7 8 38. 8 8 39. 3 0 0... 5 3 6 0 In Eercises 3 8, use a graphing utilit to solve the sstem of equations. Find the solution accurate to two decimal places. 3. e. e 5. 6. 0 8 log 0 3 6 7 0 ln 3 9 7. 8. 69 8 0 0 0 5 6 3 0 0 5 8 3 8 0 In Eercises 9 60, solve the sstem graphicall or algebraicall. Eplain our choice of method. 9. 50. 5. 3 7 6 0 5. 5 0 53. 5. 3 0 55. 56. e ln 3 57. 58. 59. 0 60. 7 0 Break-Even Analsis In Eercises 6 and 6, find the sales necessar to break even R C for the cost C of producing units and the revenue R obtained b selling units. (Round to the nearest whole unit.) 6. C 8650 50,000, 6. C 5.5 0,000, 63. Break-Even Analsis A small software compan invests $6,000 to produce a software package that will sell for $55.95. Each unit can be produced for $35.5. (a) How man units must be sold to break even? (b) How man units must be sold to make a profit of $60,000? 6. Break-Even Analsis A small fast-food restaurant invests $5000 to produce a new food item that will sell for $3.9. Each item can be produced for $.6. (a) How man items must be sold to break even? (b) How man items must be sold to make a profit of $8500? 65. DVD Rentals The weekl rentals for a newl released DVD of an animated film at a local video store decreased each week. At the same time, the weekl rentals for a newl released DVD of a horror film increased each week. Models that approimate the weekl rentals R for each DVD are R 360 R 8 R 9950 R 3.9 e 0 3 3 Animated film Horror film where represents the number of weeks each DVD was in the store, with corresponding to the first week. (a) After how man weeks will the rentals for the two movies be equal? (b) Use a table to solve the sstem of equations numericall. Compare our result with that of part (a).
3330_070.qd /5/05 9:39 AM Page 505 Section 7. Linear and Nonlinear Sstems of Equations 505 66. CD Sales The total weekl sales for a newl released rock CD increased each week. At the same time, the total weekl sales for a newl released rap CD decreased each week. Models that approimate the total weekl sales S (in thousands of units) for each CD are S 5 00 S 50 75 where represents the number of weeks each CD was in stores, with 0 corresponding to the CD sales on the da each CD was first released in stores. (a) After how man weeks will the sales for the two CDs be equal? (b) Use a table to solve the sstem of equations numericall. Compare our result with that of part (a). 67. Choice of Two Jobs You are offered two jobs selling dental supplies. One compan offers a straight commission of 6% of sales. The other compan offers a salar of $350 per week plus 3% of sales. How much would ou have to sell in a week in order to make the straight commission offer better? 68. Suppl and Demand The suppl and demand curves for a business dealing with wheat are Suppl: Demand: Rock CD Rap CD p.5 0.000 p.388 0.007 where p is the price in dollars per bushel and is the quantit in bushels per da. Use a graphing utilit to graph the suppl and demand equations and find the market equilibrium. (The market equilibrium is the point of intersection of the graphs for > 0. ) 69. Investment Portfolio A total of $5,000 is invested in two funds paing 6% and 8.5% simple interest. (The 6% investment has a lower risk.) The investor wants a earl interest income of $000 from the two investments. (a) Write a sstem of equations in which one equation represents the total amount invested and the other equation represents the $000 required in interest. Let and represent the amounts invested at 6% and 8.5%, respectivel. (b) Use a graphing utilit to graph the two equations in the same viewing window. As the amount invested at 6% increases, how does the amount invested at 8.5% change? How does the amount of interest income change? Eplain. (c) What amount should be invested at 6% to meet the requirement of $000 per ear in interest? 70. Log Volume You are offered two different rules for estimating the number of board feet in a 6-foot log. (A board foot is a unit of measure for lumber equal to a board foot square and inch thick.) The first rule is the Dole Log Rule and is modeled b V D, 5 D 0 and the other is the Scribner Log Rule and is modeled b V 0.79D D, 5 D 0 where D is the diameter (in inches) of the log and V is its volume (in board feet). (a) Use a graphing utilit to graph the two log rules in the same viewing window. (b) For what diameter do the two scales agree? (c) You are selling large logs b the board foot. Which scale would ou use? Eplain our reasoning. Model It 7. Data Analsis: Renewable Energ The table shows the consumption C (in trillions of Btus) of solar energ and wind energ in the United States from 998 to 003. (Source: Energ Information Administration) Year Solar, C Wind, C 998 70 3 999 69 6 000 66 57 00 65 68 00 6 05 003 63 08 (a) Use the regression feature of a graphing utilit to find a quadratic model for the solar energ consumption data and a linear model for the wind energ consumption data. Let t represent the ear, with t 8 corresponding to 998. (b) Use a graphing utilit to graph the data and the two models in the same viewing window. (c) Use the graph from part (b) to approimate the point of intersection of the graphs of the models. Interpret our answer in the contet of the problem. (d) Approimate the point of intersection of the graphs of the models algebraicall. (e) Compare our results from parts (c) and (d). (f) Use our school s librar, the Internet, or some other reference source to research the advantages and disadvantages of using renewable energ.
3330_070.qd /5/05 9:39 AM Page 506 506 Chapter 7 Sstems of Equations and Inequalities 7. Data Analsis: Population The table shows the populations P (in thousands) of Alabama and Colorado from 999 to 003. (Source: U.S. Census Bureau) (a) Use the regression feature of a graphing utilit to find linear models for each set of data. Graph the models in the same viewing window. Let t represent the ear, with t 9 corresponding to 999. (b) Use our graph from part (a) to approimate when the population of Colorado eceeded the population of Alabama. (c) Verif our answer from part (b) algebraicall. Geometr In Eercises 73 76, find the dimensions of the rectangle meeting the specified conditions. 73. The perimeter is 30 meters and the length is 3 meters greater than the width. 7. The perimeter is 80 centimeters and the width is 0 centimeters less than the length. 75. The perimeter is inches and the width is three-fourths the length. 76. The perimeter is 0 feet and the length is times the width. 77. Geometr What are the dimensions of a rectangular tract of land if its perimeter is 0 kilometers and its area is 96 square kilometers? 78. Geometr What are the dimensions of an isosceles right triangle with a two-inch hpotenuse and an area of square inch? Snthesis Year Alabama, P Colorado, P 999 30 6 000 7 30 00 66 9 00 79 50 003 50 55 True or False? In Eercises 79 and 80, determine whether the statement is true or false. Justif our answer. 79. In order to solve a sstem of equations b substitution, ou must alwas solve for in one of the two equations and then back-substitute. 80. If a sstem consists of a parabola and a circle, then the sstem can have at most two solutions. 8. Writing List and eplain the steps used to solve a sstem of equations b the method of substitution. 8. Think About It When solving a sstem of equations b substitution, how do ou recognize that the sstem has no solution? 83. Eploration Find an equation of a line whose graph intersects the graph of the parabola at (a) two points, (b) one point, and (c) no points. (There is more than one correct answer.) 8. Conjecture Consider the sstem of equations (a) Use a graphing utilit to graph the sstem for b,, 3, and. (b) For a fied even value of b >, make a conjecture about the number of points of intersection of the graphs in part (a). Skills Review In Eercises 85 90, find the general form of the equation of the line passing through the two points. 85., 7, 5, 5 86. 3.5,, 0, 6 87. 6, 3, 0, 3 88.,,, 5 89. 3 5, 0,, 6 90. In Eercises 9 9, find the domain of the function and identif an horizontal or vertical asmptotes. 9. 9. 93. b b. 7 3, 8, 5, f 5 6 f 7 3 f 6 9. f 3