CHAPTER 2 LINEAR EQUATIONS 2.1 CHARACTERISTICS OF LINEAR EQUATIONS 22 GRAPHICAL CHARACTERISTICS 23 SLOPEANTERCEPT FORM 2A DETERMINING THE EQUATION OF A STRAIGHT LINE 25 LINEAR EQUATIONS INVOLVING MORE THAN TWO, VARIABLES 2.6 ADDITIONAL APPLICATIONS IMPORTANT FORMULAS ADDITIONAL EXERCISES, CHAPTER TESTCHAPTER OBJECTIV Provide a thurough understanding of the algebraic and graphical characteristics of linear equations Gi Provide the tools which will allow one to determine the equation which represents a linear relationship Q Mlustrate a variety of applications of linear equations w z ° — x > gNOM NG Sear Rin ia rae Nuclear power accounts for an increasing share of electricity produced in the United States, Annual date gathered by the North American Electrie ity Council will be presented which estimates the percentage of electricity ‘generated by muclear pawer sources during the 1980s. Using selected data points, the council wants ta determine an estimating equation which could be used t0 forecast future contributions made by nuclear power sources (EXam- ple 19).- ‘The study of linear mathemstics is significant for « number of reasons. Firat, man| real-world phenomena, which we might be interested in representing mathemati cally, either are linear or can be approximated reasonably well using linear rel tionships. Asa result, linear mathematics is widely applied. Second, the analysis linear relationships is generally easier than that of nonlinear relationahips. the methods of analyzing nonlinear relationships are sometimes similar to, of extensions of, those used in linear mathematics. Consequently, a good unders ing of linear mathematics is prerequisite to the study of nonlinear mathematics! 2.1 CHARACTERISTICS OF LINEAR EQUATIONS General Form LINEAR EQUATION WITH TWO VARIABLES: Attinear equation fnvolving Fwo variables x and y has the standard form, acthy= ec (Qa) where a, 6, and care constants and # and 4 cannot both equal zero Linear equations are first-degree equations. Each variable in the equation i raised (implicitly) to the first power. The presence of terms having exponents othe! than 1 (e.g.,x%) or of terms involving a product of variables (e.g., 2xy) would exclu tan equation from being considered ti The following are all examples of linear equations involving two variables: * Most chapters begin with » Motivating Scenario. Their purpose isto provide an exemple af the of application a student should be able to solve upon completion of the chapter2.1 CHARACTERISTICS OF LINEAR EQUATIONS (Note: The names of the variables may be different from x and y,) ‘The following are examples of equations which are not linear. Can you explain why? Qe+ Bry—4y = 10 xtyi=6 vu+vo=—10 might not appear to be linear. However, multiplying both sides of the equation by 4 and moving all variables to the left side yields 3x + 2y = 40, which is in the form of Eq, (2.1) Representation Using Linear Equations Given a linear equation having the form ax + by =, the solution set for the equation is the set of all ordered pairs (x, y) which satisfy the equation. Using set notation the solution wet S can be specified aa Sala Mart e (2.2) Verbally, this set notation states that the solution set S consists of elements (Gz, y) such that (the vertical line) the equation ax + by =c is satisfied. Stated differently, Eq, (2.2) expresses that S consists of all ordered pairs (x, y) such that ax + by —c. Forany linear equation, S consists of an infinite number of clementa; that is, there is an Infinite number of pairs of values (x, y) which satiafy « Tinear equation having the form ax + by = c. To determine any pair of values which satisfy a linear equation, assume a value for one of the variables, substitute this value into the equation, and solve for the36 Caen Dion CHAPTER 2 LINEAR EQUATIONS corresponding value of the other variable, This method assumes that both variables are included in the equation (i.e,, a * 0 and b #0). Given the equation 2e+dy = 16 (a) Determine the pair of values which satisfies the equation when x= —~2. (b) Determine the pait of values which satisfies the equation when y = 0. SOLUTION (a) Substituting x~—2 into the equation, we have 2-2) + ay = 16 4y=20 yeas When x = —2, the pair of values satisfying the equation is x (8) Substituting y= 0 into the equation gives —2and y= 6, or (-2, 5). 2x +40) = 16 216 2-8 When y = 0, the pair of values satinfying the equation is (8, 0). (Production Possibilities) A company manufactures two different products. For the coming week 120 hours of labor are available for manufacturing the two producta, Work- hhours can be allocated for production of either product. In addition, since both producta frenerate a good profit, management is interested in using all 120 hours during the week. Each unit produced of product A requires 3 hours of labor and each unit of product B requires 2.5 hour. (a) Define an equation which states that total work-hours used for producing x unite of product A and y units of product B equal 120. (0) How many unita of product A can be produced if 90 unite of product 11 are produced? (c) If management decides to produce one product only, what is the maximum quantity which can be produced of product A? The maximum of product 5? SOLUTION (a) We can define our variables aa follows ‘The desired equation has the following structure as)2a CHARACTERISTICS OF LINEAR EQUATIONS. More specifically, (2.4) Since the total bours asad in producing # product equals hours required per unit produced times number of unita produced, Bq. (2.4) reduces to Sx +25y: 120 (2.6) (0) 1f 30 unite of product # are produced, then y= 30. Therefore Sx + 2.5(90) = 120 ax= 45 xa ‘Thus, one pair of values aatinfying Eq. (2.5) is (15, 30). In other words, one combination of the two products which will fully utilize the 120 hour is 16 units of product A and 90 unita of product B, (©) If management decides to manufacture product A only, no units of product B are pro- duced, or y= 0. If y= 9, ax + 2.5(0) = 120 ax = 120 “0 ‘Therefore 40 is the maximum number of unita of product A which can be produced using the 120 hours If management decides to manufacture product B only, x= 0 and 3(0) + 2.5y = 190 or y= 4B unite We tated earlier that there is an infinite number of pairs of values (x, y) which satisfy any linear equation, In Example 2, are thero any members of the solution set which might not be realistic in terms of what the equation represents? SOLUTION In Example 2, x and y represent the number of units produced of the two producta. Since negative production ia not possible, negative values of x and y are not meaningful. There are egetive values which satialy Eq, (2.6). For instance, if y= 60, then Sx + 2.560) = 120 Sx=—30 ==10CHAPTER 2 LINEAR EQUATIONS In addition to negative valves, itis possible to have decimal or fractional values for x and| >. For example, ify = 40, 3x + 2.5(40) = 120 3x20 an Depending upon the nature of the products and how they are sold, fractional values may or may not be acceptable Q POINT FOR Give examples of types of products where only integer values would be reason- THOUGHT & able, Give an example of a product for which noninteger values are reasonable. DISCUSSION Linear Equations with Variables LINEAR EQUATION WITH n VARIABLES A linesir equation involving # variables x,x,x,, x, has the general form, ax, + ax, + ax, + > tax,=b 26) where d,, ®, i)...» @,and b are constants and not alla,,a,, Byes ve equal zero. Each of the following is an example of a linear equation: Gx, — 2x, + 6xy=0 —x, + Bz, — 42, + 5x, — 2, + 22g —80 Bay — y+ Ary + xg — 8x4 + tym Bey + 104 — 1225+ 40 1,260 Given a linear equation involving n variables, as defined by Eq. (2.6), the solution set S can be specified as (2.7) ‘As with the two-variable case, there are infinitely many elements in the solution set. An element in S in representad by a collection of values (x4, 9.5, . « + »Eq)s ‘one for each of the n variables in the equation. One way of identifying specific2.1 CHARACTERISTICS OF LINEAR EQUATIONS. 9 elements in Sisto assume values forn — 1 of the variables, eubstitute these into the equation, and solve for the value of the remaining variable. ven the equation (b) Determine al] members of the solution set which have values of 0 for three of the four riablon SOLUTION (a) Substituting the given values for x,, £5, and x, into the equation yields +=1) = (0) +, = 16 or x=15 ‘The corresponding element of the solution set is (2, —1, 0, 16). (b) If x, = xy~ x, ~0, then 2(0) +410) — (0) +e, = 18 2(0) +3(0) —x, + (0) = 16 or 10-16 Its, =, = 5,0, then 2{0) + 3, — (0) + (0) = 16 or az, = 16 ¥ and x Ix," 2,3, 2x, + 8(0) — (0) + (0) 18 or 2x, ~ 16 and x=8 ‘Therefore, the elements of the solution set which have three of the four variables equaling 0 are (0, 0, 0, 16), (0, 0, —16, 0), (0, #, 0, 0), and (8, 0, 0, 0). a40 CHAPTER 2 LINEAR EQUATIONS ——— PRACTICE EXERCISE In Example 2 (Production Possibilities), assume that a third product (product C}is also to be produced. Because of the additional product, management has authorized an additional 30 labor hours. If each unit of product C re~ quires 3.75 labor hours, (a) determine the equation which requires that all 150 labor hours be used in producing the three products and (4) determine the m uum number of units which could be produced of each product. Answer: (a) If 2= number of units produced of product C, 3x+2.5y+3.752= 150, (8) 50 tunits of A 60 units of & and 40 units of C bo Section 2.1 Follow-up Exercises ‘Determine which of the following equations are linear. 1 ~3y=0 2 V0x+6y——25 3 —Bs+ 2y = 200 4+ dy= 40 & 2x — Say + by = 10 6 Viz-3y——45 7 u-3v=20 Br/2+s/b=9 9 m/2+ (2m—In)/5=0 10 (x+2y)/3 ~ Bx/4 = 2x — By 11 40—3y= v4 12 0.0003x ~ 2.9945y = x + y— 3,2543 13 2x, -3x,+24=0 1A (x, ~ Br + 6x4 — 2x, + 54)/25 = 300 10 (x, +4 — x) =5 16 Sry — 4x, = 5x, + 24, ~ 2, + 36 17 WP Fey ty = 25 AB (2x, — 3x, + 24)/4 = (xy — 22,)/5 + 90 19 Consider the equation 8 ~ 120 as a two-variable equation having the form of Eq, (2.1). (a) Define a, b, and c, (®) What pair of values satisfies the equation wnen y= 10? (c) What pair of values satiafien the equation when x= 20? (d) Verbalize the somewhat unique nature of the solution set for this equation. 20 Rework Example 2if product A requires 2 hours per unit and product 5 requires 4 hours per unit. 21 Given the equation 4x, — 22, + 6z,~0: (a) What values satiafy the equation when x, = 2 and x, > 1? (6) Define all elements of the solution set in which the values of two variables equal 0. ‘22 Given the equation x, ~ 3x, + 4r4~ 2x, = — (a) What values satiafy the equation when x, — 20, x, ~ 6, and x,——4? (0) Determine all elements of the solution set for which the values of three variables equal zero. 23. The equation x, = 20 is one of « set of related equations involving four variables x,, x. xy, and x (a) What vatuos astinty the equation when x, 4,2, = 2, and x, (b) What values satinty the equation when x, ~ 10? (c) Determine all elements of the solution set for which the values of three variables equal zaro. 24 Product Mix A company manufactures two products, A and B, Bach unit of A requires 4 labor hours and ench unit af B requires § Jahor hours. Daily manufacturing capacity is 150 labor hours, (a) If x unita of product A andy units of product B are manufactured each day and all labor hours are to be used, determine the linear oquation that requires the use of 150 labor hours per day. 1522.1 CHARACTERISTICS OF LINEAR EQUATIONS 41 (8) Bow tanny unite of A cart he made ench day if 25 unita of B are manufactured each day? (c} How many unita of A ean be made each week if12 unite of Hare manufactured each day? (Assume a 5-day work week.) 25 Nutrition Planning A dietitian is considering three food types to be served at a meal Sho is particularly concerned with the amount of one vitamin provided at this meal. One ounce of food 1 provides & milligram of the vitarnin: an ounce of food 2 provides 24 milligrams; and, an ounce of food 3 provides 16 milligrams. The minimum daily require- ment (MDR) for the vitucin i# 120 milligrams (a) If x, equals the number of ounces of food type j served at the meal, determine the equation which ensures that the meal vetisfies the MDR exeetly. (@) If only one of the three food types is to be included in the meal, how much would have to be served (in each of the three poasible cases) to watiafy the MDR? 26 Emergency Airlift The Red Cross wants to airlift supplies into « South American country which has eaperienced an earthquake. Four types of supplies, each of which ‘would be shipped in containers, are being considered. One container ofa particular item ‘weighs 120, 300, 250, nod 600 pounds, respectively, for the four items. [the airplane to bbe used has a weight capacity of 60.0KX) pounds and x, equals the number of containers shipped of item j: (a) Dewemine the equation which ensures Chat the plane will be loaded wo ite weight capacity {b) IT it is decided to devote this plane to one supply item only, bow many containers could be shipped of each item” 27 Airlift Revisited In Exercine 26, ench container of a aupply item requires a specific volume of space. Suppose containers of the four items require 30, 60, 50, and BO cubic feat, raspactively. If the volurne capacity of the plane ix 15,000 exbie feat: (a) Determine the equation which ensures that the volume capacity of the planes filled exactly. (&) IC it is decided to devote this plane to could be shipped of each ivem if volume capacity is the only consideration? (c)_ Using the information from Exercise 26, what is the maximum number of contain- er wbich could be shipped of each item if both weight and volume are considered? Indicate in each ease whether weight or volume capacity is the constraining factor. 28 Personnel Riring A softwore conwulting firm hna received large contract to develop new airline revervation syste for a major airline. In order to fulfill the contract, new hiring of programmer analysts, senior programmer analysts, andsoftwareenginesra will be required. Each progcammer analyst position will cost $40,000 in salary and benefite Each senior programmer analyst porition will cont $50,000 and each software engineer position $60,000. The airline has burgeted $1.2 snillion per year for the new hirings. Ifx, equals the number of persons hired for job category j (where j= 1 corresponds to programmer unalyata, ete.) (a) Decermine the equation which ensures that total new hires will exactly consume the budget (b) If it were desired to spend the entire budget on one type of position, how many persons of each type could be bired? (©) If exnctly 10 programmer analysta are needed for the contract, what is the mani- mum number of senior programmer analyste that could be hired? Maximum mum- ber of software engineers? 29 Public Transportation New York City has received a federal grant of $100 million for improving public transportation, ‘The funds are to be used only for the purchase of new bues, the purchase of new subway cars, or the repaving of city wtreeta. C ¢ supply item only, how many containersCHAPTER 2 LINEAR EQUATIONS ‘estimated at $150,000 por tus, $160,000 por subway car, and $250,000 per mile for repaving. City officials want to determine different waya of epending the grant money. (a) Define the decision variablos and writs the equation which enaures cormplete ex: penditure of the federal grant (b) [fit has been determined that 100 buses and 200 new subway ears will be purchased, how many miles of city strvets can be rupuved? {e) Ifofficials wish to spend all of the grant money on one type of improvement, what the different possibilities? 30 Political Campuign A candidate for the position of governor of n midwestern state ‘has an advertising budget of $1.5 million. ‘The candidate's advisors have identified four ivertising options: newspaper advertisements, radio commercials, television commer: , and billboard advertisementa. The costs for these media options average $1,500, $52,500, $10,000, and $7,500, respactively. If x, equals the number of unite purchased of media option J: (a) Write an equation which requires total advertiaing expenditures of $1.5 million. (0) Ifit has boon determined that 100 newspaper: 9 radio ade, and 60 billboard ade will be used, how many televiaion ide can they purchase? 4e) 1660 billboard ads are to bo purchaned, what ia the maximum number of newspaper ‘ads that can be purchased? Maximuro number of radio ads? TV ads? 2.2 GRAPHICAL CHARACTERISTICS Graphing Two-Variahle Equations A linear equation involving two variables graphs aa a atraight line in two dimen- sions. To graph this type of linear equation, (1) identify and plot tha coordinates of any tice points which lie an the line, (2) conneet the wo points with a straight line, ond (3) extend the straight line in both directions us far ax necessary or desirable for your purposes. The coordinates of the two pointa are found by identifying any two members of the solution set. Each element in the solution set graphs as a point (x, y) in 2-apace, where x and y are the respective velues of the two variables. For example, if the values of x= 1 and y= 3 satiafy an equation, the graphical repre- sentation of this member of the solution set is the point (1, 3). ‘The graph of the equation 2x tdy 16 in found by first identifying any two pairs of values for x oad y which satisfy the equation. NOTE Aside from the case where the right side of the equanon equals 0, the easiest points to identity (algebraically) are those found by setting ooe variable equal 10 O and solving for the value of the other varinble. That ys, let = O and solve for the value of y, then let y= 0 and solve for the value of x. Observe that the resulting -ondered pivirs, (0, 9 and (x, 0), are points om Uie y and » aXes, respectively. Letting x = 0, the corresponding value for yin 4, and letting y ~ 0 resultsinz = 8. Thus (0, 4) and (8, 0) are two members of Ube solution wot, and their graphical fepresentation isDaas Oo 2.2 GRAPHICAL. CHARACTERISTICS, 43 Figure 2.1 Graph of the linear equation 2+ 4y = 16, indicnted by the two points in Fig. 2.1. The twopointa have been connected by astraight line, and the line has been extended in both directions, Juat na (0, 4) and (8, 0) are members of the solution set for the equation 2x + 4y = 16, the coordinates of every point lying on the line represent other members of the solution set. How many unique points are there on the line? There are infinitely many, which is entirely consistent with our earlier statement that there are an infinite number of pairs of values for x 1nd.y which satisfy any linear equation. In summary, all pairs of values (x, y) that belong to the solution set of a linaar equation are represented graphically by the pointa lying an the line represeating the equation, In Fig. 2.1, the coordinates of any point not lying on the line are not members of the solution net for 2x + 4y = 16. Graph the linear equation 4x — Ty = SOLUTION ‘This equation is an example where two different points will not be identified by setting each variable equal to 0 and solving for the remaining variable. Wateh what happens! If x= 0, 40)-Ty=0 or y= 0 4z—10)=0 or x= 0 Both cases have yielded the same point, (0, 0). Therefore, to identify a second point, « value other than zero must be assumed for one of the variables. If we let x= 7,CHAPTER 2 LINEAR EQUATIONS 4(7)-7y=0 ~ty 28 yd ‘Twomembersof the solution vet, then, are (0, 0) and (7, 4). Figure 2.2llustrates the grapbof the equation. Q —+—=+: Figure 2.2 Graph of the linear equation 4x—7y = 0. NOTE a rere sees aces 8 for -lye 0 graphs as.a hh straight rough the The roperty of this ih ‘equation is that the right Mibibac cemiacce, res : Intercepts Indesoribing the graphical appearance of a straight line, two significant attributes are the x intercept and y intercept. These can be described both graphically and algebraically. i Ee pears ee Tae tion is the here the graph of | ether tak Te ecto’ rae aso et22 GRAPHICAL CHARACTERISTICS DEFINITION: y INTERCEPT ‘The. intercept of an equation is the point where the of the crosses the y axis. The y intercept represents the pas yer ting x0, Por a two-variable linear equation there exist (except for two special cases) one x intercept and one y intercept. In Fig. 2.1, the x intercept is (8, 0), and the y intercept is (0, 4). In Fig.2.2, the xand.yintercepta both occur at the same point, the origin. The r intercept is (0, 0), and the y intercept is (0, 0). Examine both figures and verify that the x intercept represents a point having ay value of Oandthatthey intercept represents a point with x value of 0. The Equation x= k A linear equation of the form ax = c in a special cuse of Bq. (2.1) where b= 0. For this equation there ix no y term. Dividing both sides of the equation by a yields the simplified form x=cla Since c and @ are constants, we can let c/a =k and write the equation in the equivalent form = an where kis « real number. This linear equation is special in the sense that x equals k regardless of the value of y. Perhaps this is underutood more easily if Ea, (2.8) is rewritten as x+0y—=h ‘The variable y may assume any value as long as x= k. That is the only condition required by the equation. As a result, any equation of this form graphs as a vertical line crossing the x axis at x = k. Figure 2.3 illustrates two equations of this type. Note that for these equations there is an x intercept (k, 0) but no'y intercept (unless k=0). What happens when k = 0? The Equation y= k A linear equation of the form by ~ c is also a apecial case of Eq, (2-1) where a~0; ice, there in no xtarm. After both sides of the equation are divided by 6, the general reduced form of this case is (2.9) where k is a real number. This equation indicates that y equals & for any value of x. Again, we can vee this more readily if Eq. (2.9) is rewritten as46 CHAPTER 2 LINEAR EQUATIONS Figure 2.3 Graphs of sample equations of the form x= & Ort yeh ‘The variable x may assume any value as long ax y= k, Any equation of thia form graphs as a horizontal line crossing the y axis at y = k, Figure 2.4 illustrates two auch equations. Note that equations of this form have no x intercepts (unleas k = 0). What happens when k= 0? Figure 2.4 Graphs of sample equations of the form y= k.22 GRAPHICAL CHARACTERISTICS a7 Slope Any atraight line, with the exception of vertical lines, can be characterized by its slope. By "slope" we are referring to the inclination of a line— whether it rises or falls as you move from left to right along the x axia—and the rate at which the line rises or falls (in other words, how steep the line is). ‘The slope of a line may be positive, negative, zero, or undefined. A line with « positive slope rises from left to right, or runs uphill. For such a line the value of y increases as x increases (or conversely, y decreases as x decreases). A line having a negative slope fails from left to right, or runs downhill, For such a line the value of y decreases as x increases (or conversely y increases as x decreases). This means thot « end y behave in an inverse manner; asoneincreases, the other decreases and vice versa. A lime having a zera slope is horizontal, Asx increases or decreases, y atays constant (the special case: y = k). Vertical lines (of the form x=) havo a slope which is undefined. Since x is constant, we cannot obeerve the behavior of y ax x changes. These slope relationships are illus- trated in Fig. 2.5. Ase noses Ae x neranves Asx erenvesor is constant regards, yoroiaaies vacrenies Secrnpes,y 1em8.n ot vain ofy (r=) uw) on ” ad) Powtive slope Negative slope Zero slope ‘Slope undatines Figure 2.5 Slope conditions for straight lines The slope of a line is quantified by a real number. The.sign of the slope (number) indicates whether the line is rising or falling. The magnitude (abeclute value) of the slope indicates the relative steepness of the line. The slope tells us the rate at which the value of y changes relative to changes in the value of x. The latger the absolute value of the slope, the steeper the angle at which the line rises or falls In Fig, 2.6c lines AS and CD both have positive slopes, but the slope of CD is larger ‘than that for AB. Sicnilarly, in Pig. 2.66 lines MN and OP both have negative slopea, but OP has the larger slope in an absolute value sense; it is more steeply sloped. Given any two points which lie on a (nonvertical) straight line, the slope can be computed as # ratio of the change in the value of y while moving from one point to the other divided by the corresponding change in the value of x, or8 CHAPTER 2 LINEAR EQUATIONS J Sal powtve ‘dope a ” and Ax “the change in the value of x," The two-point formula is one way of determining the slope of a straight line connecting two points. (2.10) Figure 2.7 illustrates the computation of Ax and Ay for the line segment PQ. ‘To compute the slope of the line connecting (2, 4) and (6, 12), arbitrarily identify one point 0 (x,,y,) and the other as (x5, y,). Given the location of tha two points in Fig. 2.8, let's label (2, 4) as (x,,,) and (5, 12) a8 (x5, y4). In moving from (2, 4) to (5, 12), the change in the value of y in dy=s— ye 12-482.22 GRAPHICAL CHARACTERISTICS Figure 2.7 r Measuring Ax | and Ay. jimilarly, the corresponding change in x is dr=x, rT)CHAPTER 2. LINEAR EQUATIONS The slope is positive, indicating that the line segment rises from left to right. The sign combined with the magnitude indicate that in moving along the line wegment, y increases at ‘rate of & unita for every 3 unite that s increases | PRACTICE EXERCISE | Verify that the result from Eq. (2.10) is unaffected by the choice of (x, 14) and (x, ya). In Example 7, label (5, 12) as (x,, y\) and (2, 4) as (xg, ys) and recompute the slope | Another way of interpreting the slope is given by the following definition. DEFINITION: SLOPE ‘The slope is the change in the value of y if + increases by 1 unit According to this definition, the value of m= § indicates that if x increases by 1 unit, y will increase by } or 2 units. Observe thia with the sequence of pointa identified in Fig. 2.9. fringe : =« increases by § for each unit x increases,22 GRAPHICAL CHARACTERISTICS st NOTE Along any straight line the slope is constant: That is, if line ty said to have a slope of —2, the slope of the line segment connecting any two paints on the line will always equal~2 (Slope Undefined) We have already seen that the slope of « linear equation having the form y~ his 0, For « horizontal line, the value of y is always the same, and the numerator of the two-point formula, y, — y,, always equals 0. We also examined the other special case of « linear equation, x ~ A. We verified that any linear equation having this form graphe as 6 vertical line crosaing the x axis at x= k. The slope of any vertical line is undefined. This ean be verified by attempting to use the two-point formula to determine the slope of the line described by x= 5. If we choose the two points (x,.¥,) = (5,0) and (x,,¥4) = (8, —1), nubstitution into Eq, (2.10) gives which is not defined a Section 2.2 Follow-up Exercises In Exercises 1-20, identify the x and y intercepts for the given linear equation, 1 3x—4y-4 2-25 +6y=—20 a —x+3y—0 4 dr +2y— 36 5 -43~12 6 10: +900=0 7 x-2y=0 B bx—3y=0 9 ~8x+5y=-20 10 (x +y)/2— 3x ~ By +16 11 2x—3y=-18 +x 12-31 +4y—10— 72-29 +50 13 lby-90=0 14 (x~ 29/412 = (Be + 4y)/9 16 axt+by=t 16 cx—dy~e 17 pe=q 18 de—ey + f= gx—hy 18 —y-s 20 -e+fe—gy—h For Exercises 21-38, graph the given linear equation 21 23 25 27 3 22 -3:+ by =— 24 ~8x+3y— U4 26 dz + 3y—~—36 28 101—5y=0 29 30 Br~4y=0 a1 32 -2y——p 33 34 Br= 20 35 a7 36 my=9,m>0,9<0,2.3 CHAPTER 2 LINEAR EQUATIUNS In Eaercines 38-59, compute the slope of the line neyment connecting the owo points. Interpret the meaning of the slope in enc case. 88 (2, 8) and (—2, —8) 40 (3, 5) and (~1, 15) 42 (—2, 3) and (1, ~9) 44 (4,—3) and (10, 12) 46 (—2, 8) and (3, 8) 48 (—4, 20) and (~4, 30) 60 (0, 30) and (0,—15) 2 (G, b) and (—o, 6) 4 (d,—c) and (0, 0) 56 (3, b) and (—10, b) 89 (—3, 10) and (2, —5) 41 (10,—3) und (12, 4) 48 (6,8) and (—3, 28) 45 (8, ~24) and (5, —15) 47 (—6,—4) and (—5, 6) 49 (5, 0) and (~25, 0) BA (5, 0) and (0,—10) 53 (0, 0) and (a, b) 55 (—5, —6) and (5, 5) 57 (—a,~b) and (o,—b) 58 (a+b, c) and (a, c) 59 (c +d, —c—d) and (a+b, —a—B) SLOPE-INTERCEPT FORM From a Different Vantage Point In thin section we discuss another form of expressing linear equations. In Sec. 2.1 ‘we stated the general form of a two-variable linear equation as artby=c (2.1) Solving Eq. (2.1) for the variable y, we get or (2.11) For any linear equation the terms ¢/b and —a/bon the right side of Eq, (2.11) have special significance, provided that b ¥ 0. If x= 0, y ~c/b. Thus, the term c/b in Eq. (2.11) is the ordinate of the y intercept. Similarly, what happens to the value of y if x increases by one unit in Eq. (2.11)? The value of y changes by —a/b. Thus, —a/b ia the slope for the equation. ‘This information is obtained from any linear equation of the form of Eq, (2.1) if it can be solved for y. Equation (2.11) iscalled the alope-intercept form of «linear equation. Equation (2.11) can be generalized in simpler form: (2.12) where m represents the slope of the line and h is the y coordinate of the y intercept. ‘To illustrate this form, the equation bxty=102.3. SLOPE-NTERCEPT FORM can be rewritten in the slope-intercept form as yerbx+10 Hence, the slope is —5 and the y intercept equals (0, 10) NOTE Why did the author use the letter K instead of b un £9. (2.12)? So as not to confuse it with the b in Eq. (2.1)! Students have often seen the slope-intercept form stated asy=mx +b. PRACTICE EXERCISE Choose two points which satisfy the equation Sx+ y= 10 and venfy that the slope equals —5 using Eq. (2.10) ‘The equation y= 2x/3 can be rewritten in slope-intercept form us y=(Px+0 ‘Tho absence of the isolated constant on the right side implicitly euggesta that & = 0. The graph of this equation is a line having a slope of j and # y intercept (0, 0). ‘The special euse of « linear equation y = kis in the slopo-intercept form. To realize this, you rust recognize that this equation can be written in the form y = 0x + A. The absonce of the x term on the right ride implicitly muggeats that m~:0; i., the slope of the line having this form equala zero. We confirmed this in Sec. 2.2 when we discussed the graphical characteris- tice of this case. The y intercept is (0, k) for such equstions For the special cave x =, it in impossible to solve for the slope- intercept form of the linear equation. The variable y isnot a part of the equation. Our conclusion is that it is impossible ‘todetermine the slope and y intercept for equations having this form, Look back at Fig. 2.3 + tone whether this conclusion is consistent with our earlier findings. a Interpreting the Slope and y Intercept In many applications of linear equations, the slope and y intercept have meaningful interpretations. Take, for example, the salary equation yrar+25 where y = weekly salary, dollars and x= number of units sold during 1 week ‘The salary equation is linear and is written in slope-intercept form. Graphically, the equation is represented by the line in Fig. 2.10, which has a slope of +3 end y54 Figure 2.10 Salary function. PAV RaL es CHAPTER 2 LINEAR EQUATIONS: yaya =30e 25 intercept equal to (0, 25). Notice that this equation bas been graphed only for nonnegative values of x and y. Can you suggest why this would be appropriate? ‘Think back to the definition of slope. Since the slope represents the change in associated with a unit increase in x, the slope of +3 meana that weekly salary y| increases by $3 for each additional unit sold. The y coordinate of the y intercept] represents the value of y when x = 0, or the salary which would be earned if no units} were sold. This ie the base salary per week. Apolice department estimates that the total cost Cof owning and operating e petrol car can bbe oatinated by the linear equation C= 040x + 18,000 where C'= total cont, dollars and = number of miles driven ‘This equation is in slope-intercept form with a slope of 0.40 and C intercept (which is equivalent to they intercept) of (0, 18,000). The alope suggests that total coat increases at « tata of $0.40 for each additional mile driven. The C intaroept indicates n cost of $18,000 if the car ia driven zero miles, Q Section 2.3 Follow-up Exercises For Exercises 1-24, rewrite ench equation in elope-intercept form and determine the slope and y intercept. 1 3r—2y— 15 2 -x+by= 275 3 4z—3y—18 4 2s-Ty——21 Barty 6 2x~y—-5 7+H)2=-6 9 (3s—By)/4= B (-22+y//3—2 10 (-x+2y)/4=3s—y23, SLOPE-ANTERCEPT FORM 14 2s—(Gx—2y)/4 12 (~4 +2y)/2-10—2x 18 4x-3y=0 14 Bx+ dy = 24 16 3r—6y+10— 16 By—5x+20= 4x—-2y+6 17 2xt dy= 4x + Sy 18 ~bx +y~12—2y—5x 19 By~24=0 20 33+6=0 21 mzt+ny=p 22 me-n=0 2a e—dy=0 24 de ey—/ 25 Wonien in the Labor Force The number rf women in the labor force in expected to increase during the 1990s, but not as dramatically aa occurred during the 1970s. One forecesting consultant uses the linear equation n= 29.6 + 1.20¢ to predict the number of women betwoon the ages of 35 and 44 who will be in the labor force, fn this equation, n equals the number of women (aged 36 to 44) in the Labor force (measured in millions) and ‘equals time measured in yearn since 1981 (¢ = 0 corresponde to 1981), If n is plotted on the vortical axis {a) Graph the equation F (8) Identify the slope and y intervept (n intercept, bere) (¢) Interpret the meaning of the alope and n intercept in thin application, (2) Predict the number of women in thus age group who will be in the labor foree in 1995 In the yeer 2000. 26 The chamber of commerce for a summer resort ia trying to determine how many touria: will be vititing each season over the coming years. A marketing research firm hae estimated that the number of touriste can be predicted by the equation p = 275,000 + 7,600t, where p = number of tourists per year and t = years (measured from thie current eason), Thus ¢ = 0 identifioe the current vesson, ¢= 1 is the next weanon, ets, Ifp is plotted on the vertical axis: (@) Graph the equation (8) Identify the slope and y intercept (p intercept, here) (©) Inverpret the meaning of the slope and p intercept in thi 27 Think Moteie! C= [F ~ 4 is an equation relating temperature in Celsius anita to temperature meamured on the Fehrecheit scale. Lat C= degrees Celnius and F= hgroes Fabrenbeit; ustumo the equation is graphed with C measured on the vertical axis. (a) Identity the slope and C intercept. (b) Interpret the meaning of the slope and C intorvept for purposes of converting from Fahrenheit to Colsius temperatures. (@) Solve the equation for F and rework parta a and b if F iw plotted on the vertical axie 28 A police department belioves that the number of serious eriunes which occur each month can be estimated by the equation ©=1,200—126p ‘where c equale the number of soricus crimes expected per month and p equals the number of officers assigned to preventive patrol. [fc ia grapbed on the vertical axis: (@) Identify the slope and interpret its meaning. (®) Identify the c intercept and interpret ite meaning, (@) Tentify the p intercept and interpret ita meaning 29 The book value of e machine is expressed by the equation ¥ = 60,000 — 7,500¢ ‘where V equals the book value in dollars and tequals tha age of the machine expressed in years.CHAPTER 2 LINEAR EQUATIONS (a) Identify the t and V intercepts, (b) Interpret the meaning of the intercepts. «) Interpret the meaning of the slope. (d) Sketch the function. 30 BAT Scores One small college has observed « downward trend in the average SAT score of applicants to the college. Anelyais has resulted in the equation 9 620—4.5¢ whore s equals the average SAT score for # given year and ¢ equals time mensured in years since 1985 (t = 0) (@) Identify the ¢ and s intercepts. (@) Interpret che meaning of the intercepts. (Does your interpretation of the intercept make sense?) (€) Interpret the meaning of the slope. (2) Sketch the equation, 31 Product Mix A company produces two products. Weekly labor availability equals 150 Inbor-bours. Each unit of product 1 requires 3 labor-hours and each unit of product 2 requires 4.5 labor-hours. If management wishes to use all Iabor-hours, the equation 3x +4.5y = 150 is a statement of thie requirement, where x equals the number of unite produced of product 1 and y equals the number of unita produced of product 2. Rewrite the equation in slope-intercept form and interpret the meaning of the slope and y intercept. Solve for the x intercept and interpret its meaning. 82 Portfolio Management A portfolio manager is concerned that two stocks generate an ‘annual income of $15,000 ora client, The two stocks earn annual dividends of $2.40 and $3.50 per ahare, respectively. Ifx equals the number of shares of stock 1 and y equals the niamber of shares of stock 2, the equation 2Ax + 35y= 16,000 states that total annual dividend income from the two stocks should total $15,000. Rewrite the equation in slope-intercept form and interpret the meaning of the slope and -y intercept in this application. Solve for the x intercept and interpret its meaning 2.4 DETERMINING THE EQUATION OF A STRAIGHT LINE Inthis section we show how to determine the equation for alinear relationship. The way in which you determine the equation depends upon the information available. ‘The following sections discuss different possibilities. In each instance we will be seeking the slope-intercept form. Thus, we will need to identify the slope-intercept parameters m and k. Slope and Intercept ‘The easiest situation is one in which you know the slope m and y intercept (0, k) of the line representing an equation. To determine the linear equation in this almost trivial case, simply substitute m and k into the slope-intercept form, Eq. (2.12). IfDETERMINING THE EQUATION OF A STRAIGHT LINE aa you are interested in stating the equation in the standard form of Eq, (2.1), simply rearrange the terms of the slope-intercept equation, Determine the equation of the straight line which has a slope of ~6 and a y intercept of (0, 15). SOLUTION Substituting values of m= —5 and & = 15 into Fy, (2.12) given yu bx +15 Restated in the form of Eq 2.1), an equivalent form of this equation is Sixty 15 a Slope and One Point If given the slope and one point which lies on a line, we can substitute the known slope m and coordinates of the given point into Eq, (2.12) and solve for k. Given that the slope of a straight line in~2 and one point lying on the line is (2, 8) we can substitute there valuen into Eq. (2.12), yielding (=2)(2) +k Since m= —2and k= 12, the lope-intercept equation it yun2et ie And, a8 before, we can rewrite thin equation in the equivalent form axty— 12 NOTE You may be wondering which form of the linear equation — Bq. (2,1) oF Bq: (2.12) — is the correct or preferred form. Both are correct! The preferred form depends on what you intend to do with the equation. Depending on the type of analysis to be conducted, one of these forms may be more appropriate than the other, Ifthe lope of « straight line is zero and one point lying on the line is (5, ~ 30), the equation of the line can be found by first substituting the zero slope and coordinates (5, —30) into Bq (2.12)CHAPTER 2 LINEAR EQUATIONS —30 = (0}(5) + or —30=k Since m = 0 and k= —90, the wlope-intercept equation is y= 03+ (-30) or y=~30 {Point-Slope Formuls) Given s nonvertical etraight kine with slope m and containing the point (x,y), the alope of the line connecting (z,, y,) with any other point (x, yon the line ‘would be expressed as Rearranging this equation results in yoy me—)) (2,18) which is the potnt-slope formula for a straight line, This formula can be used to determine the equation of a nonvertical straight Line given the slope and one point lying om the tine ‘Suppose that # line has w slope of 5 and contains the point (~4, 10). Substituting into'Bq. (2,18) and wolving for'y, == 5r+20 ym 5x+30 hich ie the dlope-intercept form of the equation Considering the linear equation x — By = 24 (@) What is the slope of the live represented by the given equation? () What in the alope of wny line parallel to the given line? (c) What io the slope of any line perpendiculur to the given Tine? (d) How many different lines nre pérpendicular to this line? {e) Find the equation of the line which is perpendicular to the given line and which panson through the point (2,5) ‘SOLUTION (a) ‘The given equntion enn be restated in alope-intercept form as —by= 43s or ye-4+ie Prom this equation the slope equals +4, and the y intercept occurs at (0,24 DETERMINING THE EQUATION OF A STRAIGHT LINE oy PARALLEL LINES ‘Two lines wre parallel if they have rhe same slope. Since the slope of the given line equals +4, any parallel lines will have a elope of +4. PERPENDICULAR LINES Ifa fine has a slope'm, (mm, * 0). the slope of any line perpendicular to the tiven line has a slope equal to the neyative reciprocal of the given lime, or m= Nim Since m, = {, the slope of any line perpendicular to the line 3x ~ 6y = 24 ia (d) Because there is an infinite eet of lines with m= —2, an infinite number of tines are perpendicular to this line. (e) The line we are interested in has « slope equal to—2 and one point on the line is (2, 6). Substituting these values into the point-alope formula, Bq. (2.13), gives 2x — 2) y-S—-254+4 -ax+9 or, alternatively, axty=9 Figure 2.11 illustrates the lines in this example, Two Points ‘A more likely situation is that some data pointa which lic on a line have been gathered arid we wish to determine the equation of the line. Assume that we are given the coordinates of two points which lie on a straight line. We can determine theslope of the line by using the two-point formula (Eq. (2.10)]. As soon as we know the slope, the y intercept can be determined by using either of the two data points ‘and proceeding as we did in the previous section.60 CHAPTER 2 LINEAR EQUATIONS Figure 211 r Family ates yt parpancieiar 0 Be by = 24 To determine the equation of the straight line which passes through (—4, 2) and the origin, “as we substitute the coordinates into the two-point formula, resulting in Substituting m= —} and the coordinates (~4, 2) into Eq. (2.13) yields y-2=(-plx 4a] yrtenjr-2 york ‘Thus, the slope-intorcept form of the equation is yack NOTE — inthis last example you might have realized that the origin is the y intercept. How would this have simplified the analysis?24 DETERMINING THE EQUATION OF A STRAIGHT LINE 61 20% | 2 yn 6 ae | sh oO a8 a0 Ext re 2.12 Percentage of total electricity produced in the United States attributable to nuclear sources. (Sources: Chicago Tribune, North American Electricity Council.) PEREEED) (Xe Power Uutiention; Motivating Seenario) Figure 2:12 iatraten sample data gathered by the North Americ1n Electricity Council. ’he graph illustrates the percent age of total electricity produced in the United States generated by nuclear power sources. ‘The percentage appears to be increasing approximately at & linear rate over time. The council wants to determine a linear equation which approximates the relationship between the percentage of electricity generated by nuclear sources (p) and time (2) meaaured in years. An analyst has chosen to fit a line through the data points for 1981 and 1986. The values for p ‘were 0.124 and 0.172 for the years 1981 and 1986, respectively. In determining the estimating, jet Cequal years meamured since 1980 (i.a,.¢ = 0 corresponds to 1980), {= 1 corresponds to 1981, ete.). Using this definition, the two date points have coordinates (1, 0.124) and (8, 0.172). The slope-intercept relationship for this ‘exaxnple will have the form pmetk (2.14) Given the two data points, the slope is Using @ slightly different, but equivalent, procedure to the poi substitute m = 0.0096 and the coordinates (1, 0.124) into Eq, (2.14): salope formula, we 0.124 = (0.0096)(1) + ke O14 hk62 CHAPTER 2 LINEAR EQUATIONS ‘Therefore, the slope-intercept form of the estimating equation ie p= 0.0006 + 0.1144 (Nuclear Power Utilization, continued) In the last example: (a) Interpret the meaning of the slope and p intercept. (b) According to this estimating equation, what parcentage of electricity ix expected from nuclear sources in the yeur 2000? {e) According to this equation, when will the percentage surpass 25 percent? SOLUTION (a) The slope indicates that for each additional year, the percentage of electricity attributa- ble to nuclear sources increases by 0.0096, or by 0,96 percent. The p intercept indicates that the entimated percentage for the year 1980 was 0.1144, or 11.44 percent. (8) A ¢ value of 20 corresponds to the year 2000. Substituting this into the estimating equation gives p= 0,0096(20) + 0.1144 0.192 + 0.1144 = 0.3064 ‘This estimating equation predicts that 20.64 percent will be attributable to nuclear sources in the year 2000. (c} Letting p = 0.25 yields (0.25 = 0.00060 + 0.1144 0.1356 = 0.00962 0.1356 t 0.0096 M15 = ‘Tharefors, when {= 14.125, the percentage will equal 0.25. Therefore, the percentage will surpass 25 percent sometime during the fifteenth year, or during 1905. NOTE Estimation plays a very iinportant part in applying mathematical analysis to the world around us Although the tools of matkiematical analvsis are most often Nery precise, the relationships which we analyze are not always exact, There are ‘mhatry applications in which the marhematical:relationships are deterrained pre- sisely. However, we offen must estimate the relationships which exist between ‘variables that are of interest. There are scientific procedures which cam be used try develop our estiniates. The use of such procedures onhances the likelihood Unat ‘bur estimates are reasonable. However, as a person actually conducting mathe matical analysis or ¥8'8 person who is the recipient of the results of such analyses, one should be aware that estimated relationships are usually. accompanied by24 DETERMINING THE EQUATION OF A STRAIGHT LINE 63 some t2eaure of error An Ateimpt should be made to understand the uf poleutial error whica is associated with estimates and to consider the effects of such error in drawing conclusions trom the mathematical analysis. As we Move through the text, you will sometimes develop mathematical relationships and sometimes be given relationships, Try to ra a questioning attitude about the source of cach relationship, Be euros about their Anticipate the umplications of crrors in the relationships, The author will try to reinforce this perspective Section 2.4 Follow-up Exercises #1 ~ 26, determine the alope intercept form of the linear equation, gives the liste’ 1 Slope =~ 2. intercept = (0, 10) 2B Slope = 4, y intercept = (0, ~5) 3 Slope ~ j, y intercept = (0, 4 Slope = ~¢..y intercept = (0, ~20) '§ Slope = ~r, y intercept = (0, ~t/2) @ Slope undefined, infinite number of y intercepts 7 Slope = —3, (4, ~2) lies oa line @ Slope = 5, (~3, 12) ties on lipe Slope = $, (~5, ~8) lies on line 10 Slope = ~4, (~4, 0) lies on line 11 Slope = 2.5, (~2, 9) ies on line 12 Slope = ~3.25, (1.5, ~7.5) lies on line 13 Slope = 6.6, (2.4, ~4.8) lies on line 14 Slope = ~8.2, (—0.75, 16.8) lies on line 16 Slope = w (p, 4) ties on line 16 Slope ~~ a, (4, ~4) lies on line 17 Slope undefined, (—3,—5) lies on line 18 Slope = 0, (20, —10) lies on line 19 Slope = 0, (u, o) lies on line 20 Slope undefined, (—t, v) lies on line 21 (4, 5) and (~2, —3) lie on line 22 (3, —2) and (12, 1) lie on line 23 (20, 240) and (15, 450) lie on line 24 (— 12, 760) and (8, ~ 1,320) lie on line 26 (0.234, 20.76) and (2.642, 18.24) liv on line 26 (5.76, ~2.48) and (3.74, 8.76) lie on line 27 (a, 6) and (c, d) lie on line 2B (a, ~3) and (o, 15) lie on line 29 (4, b) and (e, b) lie on line 30 (p,) and (—p, r) lie on line 31 Passes through (2, —4) and is parallel to the line 3x — 4y = 20 82 Passes through (~2, 10) and is parallel to the line bs ~y = 0 33 Passes through (7, 2) and is parallel to the line (a) x= 7 and (b) y=6 ‘84 Basten through (20, ~30) and ie perpendicular to the line 4x + 2y =~ 18 ‘36 Paoses through (~8, —4) and is perpendicular to the line 8x ~ 2y =0 84 Passes through (7. 2) and is perpendicular to the line (a) «= 7 and (b) y= 6CHAPTER 2 LINEAR EQUATIONS 87 Depreciation The value of a machine is expected to dcereaae at linear rate over time ‘Two data points indicate that the valve of the machine at ¢= 0 (time of purchase) is 318,000 and its valve in 1 year will equal $14,500. (a) Determine the alope-intercept equation (V = mé +k) which relates the value V of the machine to ite age ¢ (®) Interpret the meaning of the slope and V intercept. (€) Solve for the ¢ intercept and interpret ita meaning. 38 Deprociation The value of a machine is expected to decrense at a linear rate over time. ‘Two data pointa indicate that the value of the machine 1 year after the date of purchase will be $84,000 and ita value after 5 years is expected to be $36,000. (@) Determine the slope-intercept equation (V= mt + k) which relates the value V of the machine to ite age ¢, stated in years. (®) Interpret the meaning of the slope and V intercept. (€)_ Determine the ¢ intercept and interpret its meaning. 30 If C equals degrees Celsius and F equals degrees Fahrenheit, assume that the relation: hip between the two temperature scalow is linear and is being graphed with F on the vertical axis. Two data points oa the line relating C and F are (5, 41) and (25, 77). Using these points, determine the slope-intercept equation which allows transformatioa from Celsius temperature to equivalent Fahrenheit temperature, Identify and interpret the meaning of the slope, C intercept, and F intercept 40 College Retirement The largest retirement program for college professors is the Teachera Inaurance and Annuity Association/College Retirement Equities Fund (TIAA/CREF), One of the investinent options in this program is the CREF Money Market Account, which was initiated in 1988, Figure 2.13 illustrates the performance of this investment during the first 10 quarters ofits existence, Note that Vis the value of a share (unit) in this fund and that the data points reffect the end-of-month values. It ‘appaars that the value of this money market baa been increasing at an approximately linear rate. If the data points (6, 11.32) and (9, 12.04) are chosen to estimate the rela tionship between the value of a share V and time f, metsured in quartera aince the inception of the CREF Money Market Fund (¢ = 0 corresponda to March 31, 1988): CREF money market account per share (unit) quarter-end values, “June apt, Dee, Mar, June Sept Dwr, Mat, June Sept Tend of month) 1088 38 SE 89 89 HO BD 9D WM WO 2304 7 8 024 DETERMINING THE EQUATION OF A STRAIGHT LINE (a) Determine the slope-inteecept form of the estimating equation. 4b)" Identify and interpret the meaning of the slope {e) Forecast the value per share on June 30, 1991, and March 81, 1992, 41 College Retirement (continued) The CREF Money Market Fund waa established ‘on March 81, 1989. The initial value per abare was vet at $10.00. (a) Uning the equation found in part a of the previous exercise, estimate the value per hare on March 31, 1988. How much error in there in the estimate? 8) Similarly, determine the actual values per share on June 30, 1991, and March 31, 1992,* and compare with the forecasta in part cof the previous exercise. How much error was there? {c) While you have access to the data in part b, test the accuracy of the estimating equation for other quarterly date pointa 0,908) | a, ais ee tes _ a id fee F Figure 2.14 Percentage of high school students who have ssed marijuana in the past 30 days. 42 Marijuana Usage among High School Students Figure 2.14 illustrates some sur- vey data regarding the usage of marijuana among high school students, A sample of high school students was talcen every 2 years between 1979 and 1989. The data in Figure 2.14 reflects the percentage of students surveyed who indicated they bad used marijuana during the previous 20 days. The data points ®uggest that the percentage of students having used marijuana is decreasing at an approximately linear rate over time. If the * Contact the Teachers [naurance and Annuity Amociation/College Retirement Equities Fund, 720 ‘Third Avenue, New York, New York 10017 (try calling 1-800-842-2783),66 Figure 2.15 ‘Coordinate axis system in three dimensions. CHAPTER 2 LINEAR EQUATIONS ata pointe for 1979 (1, 36.5) and 198911, 16.5) are used vo estimate the linear equation| which relates the percentage of atudenta P to time ¢ (t= 1 corresponding to 1979): (a) Determine the slope-intercept farm of the estimating equation. (®) Forecast the expected percentage or 1991 and 1998. (©) Interpret the meaning of the lope and P intercept. LINEAR EQUATIONS INVOLVING MORE THAN TWO VARIABLES When linear equations involve more than two variables, the algebraic properties remain basically the same but the visual or graphicel characteristics change con- siderably or are lost altogether. ‘Three-Dimensional Coordinate Systems ‘Three-dimensionsl space can be described by using a three-dimensional coor- dinate ayatem. In three dimensions we use three coordinate axes which are all perpendicular to one another, intersecting at their respective zero points. Figure 2.16 illustrates a set of axen which are labeled by the variables x; x point of intersection of the three axes is referred to as the origin. component coordinates (ordered triples), (z,, x,. 3), the coordinates of the origin are (0, 0, 0). Observe that graphing three dimensions on paper (two dimensions) requires 9 certain perspective that may be difficult to see at first. We might have drawn Fig. 2.15 auch that we were looking right: “down the barrel" of the x, axis. In that case we ‘would have no sense of depth or location relative to the x, axis. Therefore, we rotate the coordinate axes by turning the x3 axis clockwise. This allows us to havea sense of depth when the x, axis is drawn at an angle. two dimensions divide 2-space into quadrants the axes in three dimensions divide 3-space into octante. This is illustrated in FisLINEAR EQUATIONS INVOLVING MORE THAN TWO VARIABLES. a >0 neo non <0 Pret) 2.16. Note the sign characteristics i, each octant. The three-component coordi nates allow for specifying the location or address of any point in three dimensions. ‘Aswith two-dimensional coordinates, each component of (x,, x2, x3) specifies ‘the location of a point relative to each axis. Carefully examine Fig. 2.17. In order to assist in understanding this figure, a rectangular polyhedron has been sketched. Along with several other points, we are interested in the locations of the corner points of this polyhedron, Obviously, G is located at the origin, having coordinates (0, 0, 0). Point F ties directly on the x, axig, 4 unite out. Ite coordinates are (0, 4, 0). Point A forms the upper left corner of one end (ABGH) of the polyhe- dron. Since H lies on the x, axis and A is vertically above H, we can conclude that the x, coordinate of A is 5 and the x, coordinate of A is 0. Finally, points A, B, C, and Dall seem to be at the same height (relative to the x, axis), Because point Blies68 Panne CHAPTER 2 LINEAR EQUATIONS on the x, axis at a height of 4, we conclude that A has the same x, coordinate. Thi A is located at (—5, 0, 4). See if you agree with the coordinates of J and k. Test yaur skills and define the coordinates of points B, C, D, B, and J. Ansver as EXERCISE AOD, AS CUO, 4, 4), DIS, A, 8) RB, 4,0}, 400, 4,0), Equations Involving Three Variables Linear equations having the form yx; + ogxg tayxy— 8 graph as planes in three dimensions. Tae number of variables in an determines the number of dimensions required to graphically represent the equatior ‘Three variables require three dimensions. {t is not so important that you actuall be able to graph in three dimensions. It is more important that (1) you are able recognize & linear equation involving three variables, (2) you are aware that line equations involving three variables graph as planes in three dimensions, (3) y know what a plane is, and (4) you have some feeling for ow planes can be repre sented graphically. A plane, of course, is a flat surface like the ceiling, walls, floor of the room in which you are currently sitting or lying. Instead of the points needed to graph a line, three points are necessary to define a plane. The t! pointe must not be collinear; i, they must not lie on the same line. Take, fo example, the equation 2x, + 4x, + Bx, = 12 (2.15)} If we can identify three members of the solution set for this cquation, they will specify the coordinates of three pointa lying on the plane. Three members which are identified easily are the intercepts. ‘Thege are found by setting any two of the three| variables equal to 0 and solving for the remaining variable. Verify that when} X, = xq = 0,x5 = 4,0r(0, 0, 4) is a member of the solution set. Similarly, verify that| (6,0, 0) and (0, 3, 0) are members of the solution set and thus are pointa lying on the plane representing Eq. (2.15). Figure 2.18 shows these points and a portion of| the plane which contains them ‘When graphing equations involving two variables, we identified two points and connected them with a straight line. However, we aaw that in order to represent all members of the solution set, the line must extend an infinite distance in each direction. The same is true with the solution set for three-variable equations. To| represent all members of the solution aet for the aquation 2x, +42, + 3x, — 12, the plane in Fig. 2.18 must extend an infinite distance in all directions. FI Greph the linenr equation 2, = 0 in three dimensions, SOLUTION In this problem we are being asked to graph the solution set25 R FQUATIONS INVOLVING MORE THAN TWO VARIABLES Figure 2.18 Graph of plane representing the linear equation 2x + Ax, + 3x, — 12. 8 = (x12) a1le 70) In order to graph the equation, we ugain need to identify three noncollinear points which satisfy the equation. We see that as long as x, ~ 0, x, and x, can equal any values. For example, (0, 0, 0), (0, 2,0), and (0, 0, 4) all satisfy the equation, Figure 2.19 illustrates the graph of the equation. The equation x, = O graphs asa plane perpendicular to the x, axis and passing through x, = 0. This ia the x,+, plane (the plane which includea among ita points all points lying on the x, axis and the x, ais). Q Any equation of the form x, = k graphs in 3-space as a plane perpendicular to the x, axis, intersecting il at x, = k Any equation of the form x,=& where j=1. 2, or 3, will graph as @ plane Which is perpendicular to tie x, axis at & = k. Fioures 2.20 to 2.22 illustrate this property, Equations {uvolveng more than Three variables When more than three variables exist (n> 3), graphing requires more than three dimensions. Even though we cannot envision the graphical representation of such equations, the term hyperplane is ured to describe the (“would-be”) geometric 697 CHAPTER 2 LINEAR EQUATIONS Figure 219 ‘The plane x; Figure 2.20 ” Planes of the | form 3, = ke25. LINEAR EQUATIONS INVOLVING MORE THAN TWO VARIABLES Figure 2.21 7s: Planes of the form 2, = k. Figure 2.22 Planes of the form x, = k. representation of the equation, Mathematicians would, for instance, way that the equation ay txybay b= 10 is represented by a hyperplane in 4-space or four dimensions. Or, in general, an equation of the formCHAPTER 2 LINEAR EQUATIONS 0,2, +a,x5+ +> + +a,z,=b where n> 3, would be represented by # hyperplane in n-epace. Section 2.5 Follow-up Exercises 1 Given Fig. 2.23, determine the coordingtes of points A through J. 2 Given the equation x, ~ 2x, + 4x, = 10, deterrnine the coordinates of the x,, x, and x5 intercept, 3 Given the equation ~2x, + 3x, — x, ~19, determine the coordinates of the x,,x,,and 1, intercept 4 Sketch the plane 3x, =9. 5 Sketch the plane ~22, =~ 6 Sketch the plane x, = ~2, 7 Can you draw any general conclusions about the characteristics of planes which repre- sent linear equations involving two of the three variables? For example, the equation x, + a4 = Gdoes not contain the variable x, but can be grapbed in three dimensions. How does this equation graph? How about equations which involve x, and x,? x, and x3? 2.6 ADDITIONAL APPLICATIONS ‘The more exposure you have to word problems, the more skilled you will become in formulating them. The following examples illustrate the formulation of linear equations for different types of applications. Study thee carefully and try as many of these types of problems as you can, both at the end of this section and at the end of the chapter.ADDITIONAL. APPLICATIONS 2 Emergency Airlift The International Red Crosais making plana to airlift emergency food ‘and medical supplies into a large South American city which has recently siffered from ‘extensive flooding, Four items will be airlifted in containers to aid in the recovery from flooding. Phe four items and their respective Volumes per container are shown in the table. ‘The first plane to be sent into the aree has a volume capacity of 6,000 cubic feet. Determine ‘an equation whose solution set contains all possible combinations of the four items which will fil the plane to ite volume capacity Item Volume /Contaimer, ft Blood 20 Medical supply kits 30 Food 8 Water 6 SOLUTION “The first step in almont every word problem into define the unknowns ot variables which are to be used. It in useful to ank yourself what decisions need to be made in the problem. If these decisions can be identified, they bold the key to defining the variables In this example, the decision facing the Red Cros personnel deals with how many containers of euch item should be sent on the first plane, Since the Red Cross wishes to ship ‘as many supplies na possible on thie first plane, it ia interested in identifying the different combinations which will fil the plane to capacity (volumewise). Verbally, the equation we are seeking should have the form Volurse of eappies shipped = 6000 cai ext] ‘We can be more specific by rewriting the equation as ‘Volume of blood + volurne of medical aupply kita + volume of food + volume of water = 6,000 If we let 4 * number of containers of blood 5, * number of containers of medical supply kite x, = number of containers of food , = number of containers of water ‘the equation can be stated in ita correct mathematical form as 202, + 30x, + Sx, + 6x, = 6,000 ‘Verify that each term on the left side of the equation is formed by using the relationship of item J ‘otal volume _ (ee per seocane (Se ea of item J of item JCHAPTER 2 LINEAR EQUATIONS Investment Portfolio A local university has 85 million to invest in stocks. The board of truatoes has approved six different types of stocks in which the university may invest, The current prices per share for each type of stock are indicated in the following table, Determine the equation for which the solution vet includes all the different combinations of the aix ‘stocks which could be purchased for exactly $5 million. ‘The general form of the equation should be total dollara spent on the six atocks equala ‘$5 million, or more specifically, Total dollars apart on wiock 1 -; total dollars speust on stock 2 + +++ + total doilers spent on stock 6 = $5 million ‘The basic decision to be made concerna the number of abares of each security to be pur chased so an to expend the full $5 million. Therefore, let's generalize our varinbles ma x)= number of shares purchased of stock j where j= 1, 2, 3, 4, 5, oF 6. Uning these variables, we utate the equation as 35x, +60x, + 125x, + 1003, + 500x, + 250x, = 6,000,000 ‘Note that each term on the left side of the equation haa the form [rota datar wpeot on stock j= (price par shar}(oumbor of shares purchased) Court Scheduling A metropolitan district court sorte ite cases into three categories. Court records have enabled the court clerk Lo provide eatimates of the average number of hours required to process oach type of case. Type 1 cones average 16 hours, type 2average 8 hours, and type 3 average 4.6 bours. For the coming month 860 hours are available in the different courtroom in the building. Determine an equation whose solution set includes all the different combinations of the three types of cases which would schedule the courta to their capacity. SOLUTION ‘The general form of the equation abould be ‘Total court hours echaduled = 85)ReQuone 26 ADDITIONAL APPLICATIONS 45 Latting x, x, and £, equal the number of cases scheduled of types 1, 2, nd 3, respectively, the equation is 16x, + 8x, + 4.5x, = B50 Nutrition Planning A dietitian at a local school isplanning luncheon menus. He bas eight choices of itema which may be served at any one meal. One concern of the dietitian is meeting various nutritional requirementa. Our dietitian is interested in determining the vvarigua quantitioa of each of the eight foods which would provide exactly 45 milligrams of a required vitamin, The vitamin content per serving of each of the eight food items is showa in the table. Determine the equation whose solution set aatiafies this requirement. Food Type 12 oe 8 6h 8 mgerving 5 75 3 45 98 0 25 6 SOLUTION Letting 2; = number of servings of food j, where j = 1, 2,3, 4, 6.6, 7, of 8, the equation i Bay + 7.5x, + xy + 45x, + 9x, + 10x + 2.52, + 6x, = 45, Q Section 2.6 Follow-up Exercises 1 Ine realistic sense, what are the maximum and minimum possible quantities of each item in Example 227 2 Assume that the plane in Example 22.can only earry 40,000 pounds of cargo and that the items weigh 150, 100, 60, and 70 pounds per container, respectively, State the equation ‘hove solution vet contains all combinations of the four items which will add up to equal the weight capacity of the plane. 3 Ina realintic sense, what gre the maximom and minimum allowable values for each variable in the equation developed in Example 237 4 The expected annual dividends por share of each of the preceding stocks are shown in the following table. Assume that the board of trustees desires to earn annual dividend ‘9f $1,000,000 from its investments. Using the same variables aa in the example, develop the equation whose solution set includes all possible combinations of the six stocks which will generate annual dividends equal to $1,000,000, Stock 1 Revie el 5 Expected annual dividend "$5 $8 $4 87.5030 40 5 In which of the last four examples (Examples 22~25) ahould the variablon be restricted to inteyer values? 6 A atudent is taking five courses and is fncing the crunch of final exams. She estimates that ahe hes 40 hours available to study. If x)= the number of hours allocated to studying for course j, state the equation whooe solution eet epecifies all possible alloce tiono of time among the five courses which will exhaust the 40 houre available.76 CHAPTER 2 LINEAR EQUATIONS, 7 Product Mix A firm produces three products. Product A requires 6 hours of produc tion time, product B requires 3.5 hours, and product C requires 7.5 hours for each unit produced. If 240 hours are available during the coming week, determine the aquation whose solution set apecities all possible quantitiee of the three producte which can be produced using the 240 hours. What are the maximum quantities which could be pro- duced of each product if only one product ia produced? 8 Transportation A manufacturer distributes its product to four different wholesalers ‘The monthly capacity is 40,000 units of the product. Decisions need to be made about how many units should be shipped to exch of the wholesalers. Determine the equation ‘whooe solution set specifies the different quaatitien which might be hipped if all 40,000 ‘unite are to be distributed. 9 Advertising A national firm is beginning an advertising campaign using television, radio, and newspapers. The goal is to have 10 million people see the advertisements. Past experience indicates that for every $1,000 allocated to TV, radio, and newspaper advertising, 25,000, 18,000, and 15,000 people, respectively, will wee the advertisement. ‘The decisions that need to be made are how much money should be allocated to each form of advertising in order to reach 10 million people. Determine the equation whose ‘olution vet specifies all the different advertising allocations which will result in the achievement of this goal. Ifonly one medium is to be used, how much money would have to be spent for each medium to reach 10 million people? 10 Agricultural Planning An agricultural company has « goal of harvesting 500,000 bushels of soybeans during the coming year, The company has three farms available to ‘meet this goal. Because of climate differences and other factors, che yields per acre in the different locations are 45, 90, and 26 bushels, respectively, for farme 1, 2, and 3. The decision which needs to be made concerns how many acres should he planted in soy: beans at each farm in order to meet the company's goal. State the equation which allows for specifying the different possibilities for meeting the 500,000-Lvushel goal, Ifthe entire soybean goal is to be achieved using just one farm, how many acres would be required at each farm? Q IMPORTANT FORMULAS ax+by=e Linear equation: two variables (2.1) ax, +ayx,+'-++0,x,=b Linear equation: n variables (2.2)ADDITIONAL. EXERCISES wo YASS Two-point formula (2.10) y=mx+hk — Slope-intercept form of linear equation (2.12) yo y= mx — x) ‘oint-elope formula (2.13) Q ADDITIONAL EXERCISES Section 2.1 In Exercises 1-12, determine whether the equation is linear. Bree eewe 14 x/9-y/A=2x—y+12 (e+ ay/Bey 2Jx—Biy— 2h 02x —OBy = 10 4/2 a9 tat lx — dy) = 10+ 2/3 (x—y + 19)/8 + 5y = —3(x + 12) ry ~ dry + Bex) 5x5 ~ 100 10 + 10x~4y=—4 (xy — 62 + 545)/20 = 2x, ~ x3) WISH F i + y/2= 9-2 + By wa Accompany manufactures (wo different products, A and B. Each unit of product A costa $6 to produce, and each nit of product A conta $4. The company insista that total costa for the two products be 3500. (a) Define the cost equation which states that the total cost of producing x unite of product A and y unite of product B equala $500. () Assuming the company has agreed to fil an order for 60 unite of product A, how many units of product B should be produced if total costa are to be kept at $5007 T= FH FD A local travel agent has been authorized to sell three new vacation packages for @ major airline, The three packages are priced at $800, $950, and $1,200, respectively. The airline has promised a sizable bonus commission if total sales by the travel agent equal $100,000 or more. If xy, x,, and x, equal the number of packages rold of types 1, 2, and 3, respectively: (a) Define the equation which atates that total sales equal $100,000. (b) Ifthe airline specifies that the agent must sell 20 of the $1,200 packages and 10 of, the $960 packages in order to qualify for the bonus, how many of the $800 packages will be necessary to qualify? (c} One strategy being considered by the agent is to sponsor a charter fight in which all persons would select the same package. Given that three different charters could be planned, how many persons would have to sign up for each in order ta qualify for the bonus?CHAPTER 2 LINEAR EQUATIONS 15 Fund-Raising A local theater company is attormpting to raise $1 million for expansion of seating capacity. They have undertaken a fund-raising drive to obtain the funda ‘Their campaign is soliciting donationa in three different catagories. The “Friend” cate- gory requires « donation of $1,000; the “Patron” catepory requires a donation of $5,000; the “Sustaining Member” category requires « donation of $10,000, [f x, equale the umber of donors in category j (j = 1 for Friend”) (a) Determine the equation which ensures that donations from the three categorios equal $1 million () If the goal is to be realized with donations from only one catagory of giving, how many donors would be required in each category to provide the entire $1,000,000? Section 2.2 In Exercises 16-28, identify the x and y intercepts if thay exist and graph the equation, 16 —3r—y/2 17 x/3——4 18 (y- 4/2 ar +9 19 3x—6y=0 20 4x—2y=—10 21 2 - By + 20——5r+ 2y—8 22 5-3:+6y—-x+5—2y 23 by 2y +24 24 —6r+ 24-12 +55 26 -2x+dy—=—96 26 (x — fy)/2——3y +10 QT aty~20—0 28 (2x — 4y)/2 10+ (—x + y)/9 In Exercises 29-40, compute the alope of the line segment connecting the two points, Intarpret the meaning of the slope. 29 (6, 2) and (~10, 5) 30 (=3, 8) and (1, ~14) 31 (—b, a) and (—b, 3a) 82 (2a, 3b) and (—3a, 3b) 33 (4, —5) and (2,25) 34 (—2, 40) and (3, 75) 86 (4.38, 2.64) and (~1.24, 6.82) 96 (—16.2, 4.5) and (8.62, ~1.6) 97 (m,n) and (—m, =n) 38 (—2a, 4) and (46, ~2a)ADDITIONAL EXERCIS 39 (0, f) and (—1,0) 40 (—4,c) and (4, 5) Section 2.3 In Exercises 41 and y intercept rewrite each equation in slope-intercept form and determine the slope 4 2x—By+ 10—>—4y +22—5 AZ 3x—By= 2A + x ~ By = (6x—2y)/2 44 Sx—6y— 36+ 40 Bx—4y~ 60—3r+y 46 x/2=20-y/3 43 (x—4y)/3 AT mx—ny=p 48 axtbymctdrtey 49 90x —4y + 24 = By + 30x ~ 12 50 =cr+ eye G1 y/2+3x—10= (x +y)/2 52 x—3y=3y+6x—40 53 A local dairy association enlists the help of « marketing research firm to predict the | demand for milk The research firm finds that the local demand for milk can be pre- dicted by the equation q = ~ 4,004)p + 10,000, whare p representa the price per quart (in dollars) end g represents the number of quarte purchased per week. (a) Graph the equation (0) Identify the slope and q intercept, (c) Interpret the mening of the slope and q intercept in this application 54 A manufacturing firm has 120 hours per week available in one of its departments. Two: products are processed through this department, Product A requires 4 hours per unit and product B 6 hours per unit in this department. If x equale the number of unite of product A produced per week and y equals tbe number of unite of product 8 produced per week: (a) Determine the equation which states that total time expended for producing these two products equals 120 hours per week. (0). Rewrite this equation in slope-intercept form and identify the slope and y intercept. (c) Interpret the meaning of the slope and y intercept in thia application. 55 Starting Salaries Average starting anlaries for students majoring in businese have been increasing. The equation which predicte average starting walary in = 20,250 + 1,050¢ where « equals average starting aulary and ¢ia time meamured in years since 1990 (t = 0} (a) Tdentify the » and r intercepte for this equation. (b) Interpret thene values where they are meaningful.CHAPTER 2 LINEAR EQUATIONS 56 A company produces two producta, Each product requires a cortain amount of a raw terial, Prochict A requires 3 pounds of the raw material and product B 4 pounds. For any given week, the availability of the raw material in 2,400 pounda. If equals the ‘number of unite produced of product A and y the number of unita of product Bi (a) Determine the equation which states that total raw material used each week equala 2,400 pounds (b) Rewrite the equation in slope-intercept form and identify the slope and y intercept. (¢) Interpret the values of the slope and y intercept. 57 Capital Investments A large car rental agency is preparing to purchase new cars for the coming year. The capital budget for these purchases is $20 million. Two types of cars are to be purchased, one costing $12,000 and the other $14,600. 1f equals the number of type 1 cars purchased and y the number of type 2 cars: (a) Determine the equation which statas that the total amount spent on new purchases equals $20 milli (0) Rewrite the equation in slope-intercept form. (©) Identify the slope, y intercept, and s intarcept and interpret their meaning, Section 2.4 In Exercises 58-73, use the given information to determine the slope-intercept form of the Jinoar equation. ‘58 Slope undefined and line paases through (~3, 5) ‘5 Slope undefined and line paxses through the origin 60 Slope equals 4, ¥ intercept at (0, ~20) 61 Slope equals zer0, y intercept at (0, 6) 62 «intercept at (4, 0) and (—2, 8) lies on line 63 x intercept at (~3, 0) and (8, 64 (—3, 6) and (~1, 2) lie on line 65 (—2,—18) and (5, 24) lie on line 66 (—4, 2c) and (10, 2c) lie on line 67 (8a,—5) and (30, 10) lie on line 68 (~2.38, 10.52) and (1.52, 6.54) lie on Line 6D (24.5, ~100.6) and (16.2, 36.5) lie on line 70 Passes through (—6, 4) and is perpendicular to 3x ~ 2) = 0 71 Passes through (3, 10) and is perpendicular to 4x = 2y = ~12 72 Pansen through (—2, 8) and is parallel to ~4x + By = 20 73 Passes through (~4,~1) and is parallel to 8x ~ 2y'= 0 4) lies on line 74 An economist believes there in a linear relationship between the market price of a particular commodity and the number of units suppliers of the commodity are willing to bring to the marketplace, Two sample observations indicate that when the price equals $15 per unit, the weekly supply equals 30,000 unit; and when the price equals $20 per unit, the weekly aupply equals 48,000 unite.ADDITIONAL EXERCISES (a) Ifprice per unit, p, i plotted on the horizontal axis and the quantity supplied q in plotted on the vertical axis, determine the slope-intercept form of the equation of the line which passes through these two pointe (b) Interpret the slope of the equation in this application. (c), Predict the weekly oupply if the market price equals $25 per unit 75 High-Tech Athletic Shoes A large sporting goods retailer with multiple stores in ‘attempting to predict the demend for the latest in a seemingly never-ending stream of high-tech banketball shoer, the Nike-Bok Turbo Air-Pump. It in entimated that 300 pairs will be sold per day in the retailer's stores if the new shor is priced at $200, At a price of $175, 375 pairs are expected to be sold. (a) If price is plotted on the horizontal axis, determine the stope-intercept form of the equation for demand. (8) Predict the expected demand at.a price of $225. At a price of $160. (c) Identity the p intercept and interpret ita meaning (18 6a. a) a 40 Vs “ sop ao or Sy 7 pian eet oat po we 76 7B BOB? Ae RG 9 2 4 6 8 {0 12 w 6 we Figure 2.24 Percentage of TV households with cable television 76 Cable Television Figure 2.24 illustrates some data gathered by Nielsen Media Re. search regarding the growth of cable television. Obsarvation of the data points indicates that the growth in the percentage of TV households having cable has been approzi- mately linear. Using the data points for 1976 aud 1990 tw evtimate the linear relation- ship between the percentage P and time ¢ (¢ = 0 corresponds to 1972): (a) Determine the slope-intercept form of the linear estimating equation (b) Interpret the menning of the slope and intercept (c) Forecast the expected percentage for 1996. For the year 2000. (d) When is the percentage expected to exceed 80? 8182 CHAPTER 2 LINEAR EQUATIONS Section 2.6 ‘TT A retail store sells four products. Let x,, x4, x5, and x, represent the number of units sold, respectively, of the four products. The profita earned from each unit sold of the four producta are $12, $5, $8, and $10, respectively. Target profite for the firm are $60,000. (a) Using x,, x», x5, and x,, define an equation which states that total profit from veclling the four products equals $60,000. (0) Give the range of values (maximum and minimum) possible for each veriable in the equation developed in part a. ‘78 A woman who has recently inherited $200,000 decides to invest her inheritance in stocks, She is considering eight stocks, the prices of which are given in the following table. te WP ae ee eT Price per are $25 $50 $42.60 $35 $80 817.50 $120 $100 Determine the equation whose solution eet contains all possible combinations of the ‘sight aiocks which can be purchaned for $200,000. (Be mure to define your variables.) 79 Personnel Management The head of personne! has been given a budget allotment of $500,000 to staff an enginsering department, Four types of ermployees are needed: senior engineers at an anual salary of $60,000 each, juaior engineers at an annual salary of {$32,500 each, deafters at an annual aslary of $20,000 each, and secretaries at a salary of $15,000 each. Write an equation whose solution set contains the possible combinations of employees which could be hired for $500,000. (Be sure to define your variables.) Q CHAPTER TEST 1 Given the equation 8x — 2y =~ 48: (a) Determine the x and y intercepts. (0) Graph the equation, 2 Given the equation (x + y)/3-= 24 — x (a) Rewrite the equation in slope-intercept form, (6) Identify the slope and y intercept, {e) Interpret the meaning of the slope. 8 Given two pointe (3, 18) and (6, — 14): (a) Determine the equation of the straight line which passes through the two points. (b) Identify the slope, y intercept, and x intercept. 4 The equation P— 240,000 — 7,600¢ exprases the relationship between the estimated worldwide population P of an exotic bird which has been declared an endangered apecies and time t mensured in years since 1990 (t = 0 corresponds to 1990). Identify and interpret the meaning of the slope, P intercept, and ¢ intercept.CHAPTER TEST 5 Determine the equation of the straight line which is perpendicular to the line 3x — 2y =—28 and which passes through the point (—5, 20). 6 A manufacturer has « monthly supply of 750,000 pounds of a raw material used in making four products, The number of pounds required to manufacture a unit of each product equals 10, 15, 7.5, and 18, respectively. | andx,equal the number of units produced of each produet (a) Define the equation whose solution set includes the possible combinations of the four products which would exhaust the monthly supply of the raw material, {b) What is the maximum amount which could be made of each produet ifonly ‘one product is produced and the supply of raw material is the only limiting