94 (8 ) Chapter 8 Sstems of Linear Equations and Inequalities In this Solving a Sstem b Graphing section Independent, Inconsistent, and Dependent Equations Solving b Substitution Applications E X A M P L E calculator close-up To check Eample, graph and 4. From the CALC menu, choose intersect to have the calculator locate the point of intersection of the two lines. After choosing intersect, ou must indicate which two lines ou want to intersect and then guess the point of intersection. 0 8. SOLVING SYSTEMS BY GRAPHING AND SUBSTITUTION In Chapter 4 we studied linear equations in two variables, but we have usuall considered onl one equation at a time. In this chapter we will see problems that involve more than one equation. An collection of two or more equations is called a sstem of equations. If the equations of a sstem involve two variables, then the set of ordered pairs that satisf all of the equations is the solution set of the sstem. In this section we solve sstems of linear equations in two variables and use sstems to solve problems. Solving a Sstem b Graphing Because the graph of each linear equation is a line, points that satisf both equations lie on both lines. For some sstems these points can be found b graphing. A sstem with onl one solution Solve the sstem b graphing: 4 First write the equations in slope-intercept form: 4 Use the -intercept and the slope to graph each line. The graph of the sstem is shown in Fig. 8.. From the graph it appears that these lines intersect at (, ). To be certain, we can check that (, ) satisfies both equations. Let and in to get. Let and in 4 to get 4. Because (, ) satisfies both equations, the solution set to the sstem is (, ). 5? = + 0 0 0 = + 4 FIGURE 8. The graphs of the equations in the net eample are parallel lines, and there is no point of intersection.
8. Solving Sstems b Graphing and Substitution (8 ) 95 E X A M P L E A sstem with no solution Solve the sstem b graphing: 6 stud tip Rela and don t worr about grades. If ou are doing everthing ou can and should be doing, then there is no reason to worr. If ou are neglecting our homework and skipping class, then ou should be worried. First write each equation in slope-intercept form: 6 6 The graph of the sstem is shown in Fig. 8.. Because the two lines in Fig. 8. are parallel, there is no ordered pair that satisfies both equations. The solution set to the sstem is the empt set,. 4 = + 4 5 = 4 FIGURE 8. The equations in the net eample look different, but their graphs are the same straight line. E X A M P L E = FIGURE 8. 4 5 A sstem with infinitel man solutions Solve the sstem b graphing: ( ) 4 Write each equation in slope-intercept form: ( ) 4 4 4 Because the equations have the same slope-intercept form, the original equations are equivalent. Their graphs are the same straight line as shown in Fig. 8.. Ever point on the line satisfies both equations of the sstem. There are infinitel man points in the solution set. The solution set is (, ) 4.
96 (8 4) Chapter 8 Sstems of Linear Equations and Inequalities Independent, Inconsistent, and Dependent Equations Our first three eamples illustrate the three possible was in which two lines can be positioned in a plane. In Eample the lines intersect in a single point. In this case we sa that the equations are independent or the sstem is independent. If the two lines are parallel, as in Eample, then there is no solution to the sstem, and the equations are inconsistent or the sstem is inconsistent. If the two equations of a sstem are equivalent, as in Eample, the equations are dependent or the sstem is dependent. Figure 8.4 shows the tpes of graphs that correspond to independent, inconsistent, and dependent sstems. stud tip Correct answers often have more than one form. If our answer to an eercise doesn t agree with the one in the back of this tet, tr to determine if it is simpl a different form of the answer. For eample, and look different but the are equivalent epressions. Independent Solving b Substitution Inconsistent FIGURE 8.4 Solving a sstem b graphing is certainl limited b the accurac of the graph. If the lines intersect at a point whose coordinates are not integers, then it is difficult to determine those coordinates from the graph. The method of solving a sstem b substitution does not depend on a graph and is totall accurate. For substitution we replace a variable in one equation with an equivalent epression obtained from the other equation. Our intention in this substitution step is to eliminate a variable and to give us an equation involving onl one variable. Dependent E X A M P L E 4 An independent sstem solved b substitution Solve the sstem b substitution: 8 6 Since 6 we can replace in 8 b 6: 8 ( 6) 8 Substitute 6 for. 6 8 8 4 0 5
8. Solving Sstems b Graphing and Substitution (8 5) 97 calculator close-up To check Eample 4, graph (8 ) and 6. From the CALC menu, choose intersect to have the calculator locate the point of intersection of the two lines. 0 To find, we let 5 in the equation 6: 5 6 5 6 The net step is to check 5 and in each equation. If 5 and in 8, we get 5 () 8. If 5 and in 6, we get 5 6. 0 0 0 Because both of these equations are true, the solution set to the sstem is. The equations of this sstem are independent. 5, E X A M P L E 5 helpful hint The purpose of Eample 5 is to show what happens when ou tr to solve an inconsistent sstem b substitution. If we had first written the equations in slope-intercept form, we would have known that the lines are parallel and the solution set is the empt set. An inconsistent sstem solved b substitution Solve b substitution: 4 7 Solve the first equation for to get. Substitute for in the second equation: 4 7 ( ) 4 7 4 6 4 7 6 7 Because 6 7 is incorrect no matter what values are chosen for and, there is no solution to this sstem of equations. The equations are inconsistent. To check, we write each equation in slope-intercept form: 4 7 4 7 7 4 The graphs of these equations are parallel lines with different -intercepts. The solution set to the sstem is the empt set,. E X A M P L E 6 A dependent sstem solved b substitution Solve b substitution: 5 4 5
98 (8 6) Chapter 8 Sstems of Linear Equations and Inequalities helpful hint The purpose of Eample 6 is to show what happens when a dependent sstem is solved b substitution. If we had first written the first equation in slope-intercept form, we would have known that the equations are dependent and would not have done substitution. Substitute 5 into the first equation: ( 5) 5 4( 5) 5 5 4 0 5 5 5 5 Because the last equation is an identit, an ordered pair that satisfies 5 will also satisf 5 4. The equations of this sstem are dependent. The solution set to the sstem is the set of all points that satisf 5. We write the solution set in set notation as (, ) 5. We can verif this result b writing 5 4 in slope-intercept form: 5 4 5 4 5 5 Because this slope-intercept form is identical to the slope-intercept form of the other equation, the are two equations that look different for the same straight line. If a sstem is dependent, then an identit will result after the substitution. If the sstem is inconsistent, then an inconsistent equation will result after the substitution. The strateg for solving an independent sstem b substitution can be summarized as follows. stud tip When working a test, scan the problems and pick out the ones that are the easiest for ou. Do them first. Save the harder problems until the last. The Substitution Method. Solve one of the equations for one variable in terms of the other.. Substitute into the other equation to get an equation in one variable.. Solve for the remaining variable (if possible). 4. Insert the value just found into one of the original equations to find the value of the other variable. 5. Check the two values in both equations. Applications Man of the problems that we solved in previous chapters involved more than one unknown quantit. To solve them, we wrote epressions for all of the unknowns in terms of one variable. Now we can solve problems involving two unknowns b using two variables and writing a sstem of equations. E X A M P L E 7 Perimeter of a rectangle The length of a rectangular swimming pool is twice the width. If the perimeter is 0 feet, then what are the length and width?
8. Solving Sstems b Graphing and Substitution (8 7) 99 W L W Draw a diagram as shown in Fig. 8.5. If L represents the length and W represents the width, then we can write the following sstem. L FIGURE 8.5 L W L W 0 Since L W, we can replace L in L W 0 with W: (W) W 0 4W W 0 6W 0 W 0 So the width is 0 feet and the length is (0) or 40 feet. E X A M P L E 8 helpful hint In Chapter we would have done Eample 8 with one variable b letting represent the amount invested at 0% and 0,000 represent the amount invested at %. Tale of two investments Belinda had $0,000 to invest. She invested part of it at 0% and the remainder at %. If her income from the two investments was $60, then how much did she invest at each rate? Let be the amount invested at 0% and be the amount invested at %. We can summarize all of the given information in a table: Amount Rate Interest First investment 0% 0.0 Second investment % 0. calculator close-up To check Eample 8, graph 0,000 and (60 0.) 0.. The viewing window needs to be large enough to contain the point of intersection. Use the intersection feature to find the point of intersection. 0,000 We can write one equation about the amounts invested and another about the interest from the investments: 0,000 Total amount invested 0.0 0. 60 Total interest Solve the first equation for to get 0,000. Substitute 0,000 for in the second equation: 0.0 0. 60 0.0(0,000 ) 0. 60 Replace b 0,000. 000 0.0 0. 60 Solve for. 0.0 60 8000,000 Because 0,000 To check this answer, find 0% of $,000 and % of $8000: 0 0,000 0.0(,000),00 0.(8000) 960 0 Because $00 $960 $60 and $8000 $,000 $0,000, we can be certain that Belinda invested $,000 at 0% and $8000 at %.
400 (8 8) Chapter 8 Sstems of Linear Equations and Inequalities M A T H A T W O R K Accountants work with both people and numbers. When Maria L. Manning, an auditor at Deloitte & Touche LLP, is preparing for an audit, she studies the numbers on the balance sheet and income statement, comparing the current fiscal ear to the prior one. The purpose of an independent audit is to give investors a realistic view of a compan s finances. To determine that a compan s financial statements are fairl stated in accordance with the General Accepted Accounting Procedures (GAAP), she first interviews the comptroller to get the stor behind the numbers. Tpical questions she could ask are: How productive was our ear? Are there an new products? What was behind the big stories in the newspapers? Then Ms. Manning and members of the audit team test the financial statement in detail, closel eamining accounts relating to unusual losses or profits. Ms. Manning is responsible for both manufacturing and mutual fund companies. At a manufacturing compan, accounts receivable and inventor are two ke components of an audit. For eample, to test inventor, Ms. Manning visits a compan s warehouse and phsicall counts all the items for sale to verif a compan s assets. For a mutual fund compan the audit team pas close attention to current events, for the indirectl affect the financial industr. In Eercises 55 and 56 of this section ou will work problems that involve one aspect of cost accounting: calculating the amount of taes and bonuses paid b a compan. ACCOUNTANT WARM-UPS True or false? Eplain our answer.. The ordered pair (, ) is in the solution set to the equation 4.. The ordered pair (, ) satisfies 4 and 6.. The ordered pair (, ) satisfies 4 5 and 4 5. 4. If two distinct straight lines in the coordinate plane are not parallel, then the intersect in eactl one point. 5. The substitution method is used to eliminate a variable. 6. No ordered pair satisfies 5 and. 7. The equations 6 and 4 are independent. 8. The equations 7 and 8 are inconsistent. 9. The graphs of dependent equations are the same. 0. The graphs of independent linear equations intersect at eactl one point.
8. Solving Sstems b Graphing and Substitution (8 9) 40 8. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. How do we solve a sstem of linear equations b graphing?. How can ou determine whether a sstem has no solution b graphing?. What is the major disadvantage to solving a sstem b graphing? 4. How do we solve sstems b substitution? a) b) 4 4 c) d) 5. How can ou identif an inconsistent sstem when solving b substitution? 6. How can ou identif a dependent sstem when solving b substitution? Solve each sstem b graphing. See Eamples. 7. 8. 9. 0... 4. 4. 9 6 4 4 4 Solve each sstem b the substitution method. Determine whether the equations are independent, dependent, or inconsistent. See Eamples 4 6.. 5. 4 5 5 6. 7 4. 5 4 5. 5 6. 4 4 6 7. 5 8. 6 5 ( ) 4 6 4 5. 4 6. 6 8 The graphs of the following sstems are given in (a) through (d). Match each sstem with the correct graph. 7. 5 4 7 8. 5 9 9 5 6 8 9. 4 5 0. 4 5 4 9. 9 0. 0 5 5 4. 0. 6 5. 40 4. 0 0. 0.08.5 0. 0.05 7 5. 0 6. 5 4 5 4
40 (8 0) Chapter 8 Sstems of Linear Equations and Inequalities 7. 4 8. 5 9. 4 40. ( 4) 4. ( ) ( ) 4. 4. 0.75 44..875.5.5.5.5.85 In Eercises 45 58, write a sstem of two equations in two unknowns for each problem. Solve each sstem b substitution. See Eamples 7 and 8. 45. Perimeter of a rectangle. The length of a rectangular swimming pool is 5 feet longer than the width. If the perimeter is 8 feet, then what are the length and width? 46. Household income. Alkena and Hsu together earn $84,6 per ear. If Alkena earns $,468 more per ear than Hsu, then how much does each of them earn per ear? 47. Different interest rates. Mrs. Brighton invested $0,000 and received a total of $,00 in interest. If she invested part of the mone at 0% and the remainder at 5%, then how much did she invest at each rate? 48. Different growth rates. The combined population of Marsville and Springfield was 5,000 in 990. B 995 the population of Marsville had increased b 0%, while Springfield had increased b 9%. If the total population increased b 80 people, then what was the population of each cit in 990? 49. Finding numbers. The sum of two numbers is, and their difference is 6. Find the numbers. $84 and twice as man adult tickets as student tickets were sold, then how man of each were sold? 5. Corporate taes. According to Bruce Harrell, CPA, the amount of federal income ta for a class C corporation is deductible on the Louisiana state ta return, and the amount of state income ta for a class C corporation is deductible on the federal ta return. So for a state ta rate of 5% and a federal ta rate of 0%, we have and state ta 0.05(taable income federal ta) federal ta 0.0(taable income state ta). Find the amounts of state and federal income ta for a class C corporation that has a taable income of $00,000. 54. More taes. Use information given in Eercise 5 to find the amounts of state and federal income ta for a class C corporation that has a taable income of $00,000. Use a state ta rate of 6% and a federal ta rate of 40%. 55. Cost accounting. The problems presented in this eercise and the net are encountered in cost accounting. A compan has agreed to distribute 0% of its net income N to its emploees as a bonus; B 0.0N. If the compan has income of $0,000 before the bonus, the bonus B is deducted from the $0,000 as an epense to determine net income; N 0,000 B. Solve the sstem of two equations in N and B to find the amount of the bonus. 56. Bonus and taes. A compan has an income of $00,000 before paing taes and a bonus. The bonus B is to be 0% of the income after deducting income taes T but before deducting the bonus. So B 0.0(00,000 T ). Because the bonus is a deductible epense, the amount of income ta T at a 40% rate is 40% of the income after deducting the bonus. So T 0.40(00,000 B). 50. Finding more numbers. The sum of two numbers is 6, and their difference is 8. Find the numbers. 5. Toasters and vacations. During one week a land developer gave awa Florida vacation coupons or toasters to 00 potential customers who listened to a sales presentation. It costs the developer $6 for a toaster and $4 for a Florida vacation coupon. If his bill for prizes that week was $708, then how man of each prize did he give awa? 5. Ticket sales. Tickets for a concert were sold to adults for $ and to students for $. If the total receipts were Bonus (in thousands of dollars) 00 80 60 40 0 T 0.40(00,000 B) B 0.0(00,000 T) 0 0 0 40 60 80 00 Taes (in thousands of dollars) FIGURE FOR EXERCISE 56
8. The Addition Method (8 ) 40 a) Use the accompaning graph to estimate the values of T and B that satisf both equations. b) Solve the sstem algebraicall to find the bonus and the amount of ta. 57. Tetbook case. The accompaning graph shows the cost of producing tetbooks and the revenue from the sale of those tetbooks. a) What is the cost of producing 0,000 tetbooks? b) What is the revenue when 0,000 tetbooks are sold? c) For what number of tetbooks is the cost equal to the revenue? d) The cost of producing zero tetbooks is called the fied cost. Find the fied cost. GETTING MORE INVOLVED Amount (in millions of dollars)..0 0.8 0.6 0.4 0. 0 R 0 C 0 400,000 0 0 0 40 Number of tetbooks (in thousands) FIGURE FOR EXERCISE 57 59. Discussion. Which of the following equations is not equivalent to 6? a) 6 b) c) d) ( 5) 4 60. Discussion. Which of the following equations is inconsistent with the equation 4 8? a) b) 6 8 6 4 c) 8 d) 4 8 4 58. Free market. The function S 5000 00 and D 9500 00 epress the suppl S and the demand D, respectivel, for a popular compact disk brand as a function of its price (in dollars). a) Graph the functions on the same coordinate sstem. b) What happens to the suppl as the price increases? c) What happens to the demand as the price increases? d) The price at which suppl and demand are equal is called the equilibrium price. What is the equilibrium price? GRAPHING CALCULATOR EXERCISES 6. Solve each sstem b graphing each pair of equations on a graphing calculator and using the trace feature or intersect feature to estimate the point of intersection. Find the coordinates of the intersection to the nearest tenth. a).5 7. b). 4... 9..4 9.. In this The Addition Method section Equations Involving Fractions or Decimals Applications E X A M P L E 8. THE ADDITION METHOD In Section 8. ou used substitution to eliminate a variable in a sstem of equations. In this section we see another method for eliminating a variable in a sstem of equations. The Addition Method In the addition method we eliminate a variable b adding the equations. An independent sstem solved b addition Solve the sstem b the addition method: 5 9 4 5