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Outline

8-1 Systems of Linear Equations in Two Variables

8-2 Systems of Linear Equationsand Augmented Matrices

8-3 Gauss––Jordan Elimination

8-4 Matrix Operations

8-5 Inverse of a Square Matrix

8-6 Matrix Equations and Systemsof Linear Equations

8-7 Systems of Linear Inequalities

8-8 Linear Programming

Chapter 8 Group Activity:Modeling with Systems ofLinear Equations

Chapter 8 Review

Application

A Red Cross plane is being loaded with bottledwater and dehydrated food for transport to anearthquake region. Each bottle of water weighs 18pounds, occupies 1 cubic foot of space, and willsupply 15 people. Each package of dehydrated foodweighs 9 pounds, occupies 0.75 cubic feet of space,and will supply 11 people. The plane has space for4,500 cubic feet of cargo weighing at most 64,800pounds. How many bottles of water and packagesof dehydrated food should be loaded into theplane to maximize the total number of peoplesupplied with either food or water by thisshipment?

In this chapter we first discuss how systems of linear equations involv-ing two variables are solved graphically and algebraically. Becausethese techniques are not suitable for linear systems involving larger

numbers of equations and variables, we then turn to a differentmethod of solution involving the concept of an augmented matrix,which arises quite naturally when dealing with larger linear systems.We then study matrices and matrix operations in their own right as anew mathematical form. With these new operations added to ourmathematical toolbox, we return to systems of equations from a freshpoint of view. Finally, we discuss systems of linear inequalities and lin-ear programming. Throughout the chapter we use these new mathe-matical tools to solve a variety of interesting and important appliedproblems.

Preparing for This ChapterBefore getting started on this chapter, review the following concepts:

Properties of Real Numbers (Appendix A, Section 1)

Linear Equations and Inequalities (Appendix A, Section 8, and Chapter 2, Section 2)

Linear Functions (Chapter 2, Section 1)

Section 8-1 Systems of Linear Equations in Two Variables

Systems of EquationsGraphingSubstitutionApplications

In this section we discuss both graphical and algebraic methods for solving sys-tems of linear equations in two variables. Then we use systems of this type toconstruct and solve mathematical models for several applications.

Systems of Equations

To establish basic concepts, consider the following example. At a computer fair,student tickets cost $2 and general admission tickets cost $3. If a total of 7 tick-ets are purchased for a total cost of $18, how many of each type were purchased?

Let

x 5 Number of student tickets

y 5 Number of general admission tickets

582

8-1 Systems of Linear Equations in Two Variables 583

Then

x 1 y 5 7 Total number of tickets purchased

2x 1 3y 5 18 Total purchase cost

We now have a system of two linear equations in two variables. Thus, we can solvethis problem by finding all pairs of numbers x and y that satisfy both equations.

In general, we are interested in solving linear systems of the type

ax 1 by 5 h System of two linear equations in two variables

cx 1 dy 5 k

where x and y are variables, a, b, c, and d are real numbers called the coefficientsof x and y, and h and k are real numbers called the constant terms in the equa-tions. A pair of numbers x 5 x0 and y 5 y0 is a solution of this system if eachequation is satisfied by the pair. The set of all such pairs of numbers is called thesolution set for the system. To solvea system is to find its solution set.

Graphing

Recall that the graph of a linear equation is the line consisting of all ordered pairsthat satisfy the equation. To solve the ticket problem by graphing, we graph bothequations in the same coordinate system. The coordinates of any points that thelines have in common must be solutions to the system, since they must satisfyboth equations.

Solving a System by Graphing

Solve the ticket problem by graphing: x 1 y 5 72x 1 3y 5 18

S o l u t i o n From Figure 1 we see that

x 5 3 Student tickets

y 5 4 General admission tickets

C h e c k x 1 y 5 7 2x 1 3y 5 18

3 1 4 ‚ 7 2(3) 1 3(4) ‚ 18

7 ⁄ 7 18 ⁄ 18

Solve by graphing and check: x 2 y 5 23x 1 2y 5 23

M A T C H E D P R O B L E M

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FIGURE 1

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584 8 SYSTEMS; MATRICES

It is clear that the preceding example has exactly one solution, since the lineshave exactly one point of intersection. In general, lines in a rectangular coordi-nate system are related to each other in one of three ways, as illustrated in thenext example.

Solving Three Important Types of Systems by Graphing

Solve each of the following systems by graphing:

(A) 2x 2 3y 5 2 (B) 4x 1 6y 5 12 (C) 2x 2 3y 5 26x 1 2y 5 8 2x 1 3y 5 26 2x 1 y 5 23

S o l u t i o n s (A) (B)

Lines intersect at one point only. Lines are parallel (each hasExactly one solution: x 5 4, y 5 2 slope 2 ). No solution.

(C)

Lines coincide.Infinitely many solutions.

Solve each of the following systems by graphing:

(A) 2x 1 3y 5 12 (B) x 2 3y 5 23 (C) 2x 2 3y 5 12x 2 3y 5 23 22x 1 6y 5 12 2x 1 y 5 26

We now define some terms that can be used to describe the different types ofsolutions to systems of equations illustrated in Example 2.

32


2

23

32

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SYSTEMS OF LINEAR EQUATIONS: BASIC TERMS

A system of linear equations is consistent if it has one or more solutionsand inconsistent if no solutions exist. Furthermore, a consistent system issaid to be independent if it has exactly one solution (often referred to asthe unique solution) and dependent if it has more than one solution.

Referring to the three systems in Example 2, the system in part A is consis-tent and independent, with the unique solution x 5 4 and y 5 2. The system inpart B is inconsistent, with no solution. And the system in part C is consistentand dependent, with an infinite number of solutions: all the points on the twocoinciding lines.

Can a consistent and dependent system have exactly two solutions?Exactly three solutions? Explain.

By geometrically interpreting a system of two linear equations in two vari-ables, we gain useful information about what to expect in the way of solutions tothe system. In general, any two lines in a rectangular coordinate plane must inter-sect in exactly one point, be parallel, or coincide (have identical graphs). Thus,the systems in Example 2 illustrate the only three possible types of solutions forsystems of two linear equations in two variables. These ideas are summarized inTheorem 1.

POSSIBLE SOLUTIONS TO A LINEAR SYSTEM

The linear system

ax 1 by 5 h

cx 1 dy 5 k

must have

1. Exactly one solution Consistent and independent

or

2. No solution Inconsistent

or

3. Infinitely many solutions Consistent and dependent

There are no other possibilities.


One drawback of finding a solution by graphing is the inaccuracy of hand-drawn graphs. Graphic solutions performed on a graphing utility, however, pro-vide both a useful geometric interpretation and an accurate approximation of thesolution to a system of linear equations in two variables.

Solving a System Using a Graphing Utility

Solve to two decimal places using a graphing utility: 5x 2 3y 5 132x 1 4y 5 15

S o l u t i o n First solve each equation for y:

Next, enter each equation in a graphing utility [Fig. 2(a)], graph in an appropri-ate viewing window, and approximate the intersection point [Fig. 2(b)].

(a) Equation definitions (b) Intersection point (c) Check

Rounding the values in Figure 2(b) to two decimal places, we see that thesolution is

x 5 3.73 and y 5 1.88 or (3.73, 1.88)

Figure 2(c) shows a check of this solution.

Solve to two decimal places using a graphing utility: 2x 2 5y 5 2254x 1 3y 5 5

Remark In the solution to Example 3, you might wonder why we checked a solution pro-duced by a graphing utility. After all, we don’t expect a graphing utility to makean error. But the equations in the original system and the equations entered inFigure 2(a) are not identical. We might have made an error when solving the orig-inal equations for y. The check in Figure 2(c) eliminates this possibility.

Graphic methods help us visualize a system and its solutions, frequently revealrelationships that might otherwise be hidden, and, with the assistance of a graph-ing utility, provide very accurate approximations to solutions.


3

FIGURE 2

y 5 20.5x 1 3.75 y 5 53 x 2 13

3

4y 5 22x 1 15 23y 5 25x 1 13

2x 1 4y 5 15 5x 2 3y 5 13

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Substitution

There are a number of different algebraic techniques that can also be used to solvesystems of linear equations in two variables. One of the simplest is the substitu-tion method.To solve a system by substitution, we first choose one of the twoequations in a system and solve for one variable in terms of the other. (We makea choice that avoids fractions, if possible.) Then we substitute the result in theother equation and solve the resulting linear equation in one variable. Finally, wesubstitute this result back into the expression obtained in the first step to find thesecond variable. We return to the ticket problem stated at the beginning of thesection to illustrate this process.

Solving a System by Substitution

Use substitution to solve the ticket problem:x 1 y 5 72x 1 3y 5 18

S o l u t i o n Solve either equation for one variable and substitute into the remaining equation.We choose to solve the first equation for y in terms of x:

Solve the first equation for y in terms of x.

Substitute into the second equation.

Now, replace x with 3 in y 5 7 2 x:

y 5 7 2 x

y 5 7 2 3

y 5 4

Thus the solution is 3 student tickets and 4 general admission tickets.

C h e c k

18 ⁄ 18 7 ⁄ 7

2(3) 1 3(4) 0 18 3 1 4 ‚ 7

2x 1 3y 5 18 x 1 y 5 7

x 5 3

2x 5 23

2x 1 21 2 3x 5 18

2x 1 3(7 2 x) 5 18

2x 1 3y 5 18

y 5 7 2 x

x 1 y 5 7

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4

2


Solve by substitution and check: x 2 y 5 23x 1 2y 5 23

Solving a System by Substitution

Solve by substitution and check: 2x 2 3y 5 73x 2 y 5 7

S o l u t i o n To avoid fractions, we choose to solve the second equation for y:

Solve for y in terms of x.

Substitute into first equation.

First equation

Solve for x.

Substitute x 5 2 in y 5 3x 2 7.

Thus, the solution is x 5 2 and y 5 21.

C h e c k

Solve by substitution and check: 3x 2 4y 5 182x 1 y 5 1

Use substitution to solve each of the following systems. Discuss thenature of the solution sets you obtain.

x 1 3y 5 4 x 1 3y 5 4

2x 1 6y 5 7 2x 1 6y 5 8


5

7 ⁄ 7 7 ⁄ 7

3(2) 2 (21) 0 7 2(2) 2 3(21) ‚ 7

3x 2 y 5 7 2x 2 3y 5 7

y 5 21

y 5 3(2) 2 7

y 5 3x 2 7

x 5 2

27x 5 214

2x 2 9x 1 21 5 7

2x 2 3(3x 2 7) 5 7

2x 2 3y 5 7

y 5 3x 2 7

2y 5 23x 1 7

3x 2 y 5 7

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Applications

The following examples illustrate the use of systems of linear equations to con-struct models for applied problems. Each model can be solved by either graph-ing or substitution—the choice is really a matter of personal preference.

Food Processing

A food manufacturer produces regular and lite smoked sausages. A regularsausage is 72% pork and 28% turkey and a lite sausage is 22% pork and 78%turkey. The company has just received a shipment of 2,000 pounds of porkand 2,000 pounds of turkey. How many pounds of each type of sausage shouldbe produced to use all the meat in this shipment?

S o l u t i o n First we define the relevant variables:

x 5 Pounds of regular sausage

y 5 Pounds of lite sausage

Next we summarize the given information in Table 1. It is convenient to organizethe table so that the quantities represented by variables correspond to columns inthe table (rather than to rows), as shown.

Regular Sausage Lite Sausage Total

Pork 72% 22% 2,000

Turkey 28% 78% 2,000

Now we use the information in the table to form equations involving x and y:

We will solve this system graphically. Figure 3(a) shows the equations after theyhave been solved for y and entered in the equation editor of a graphing utility.From Figure 3(b), we conclude that producing 2,240 pounds of regular sausageand 1,760 pounds of lite sausage will use all the available pork and turkey.

T A B L E 1

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(a) (b)

A food manufacturer produces regular and deluxe rice mixtures by mixing wildrice with long-grain rice. The regular rice mixture is 5% wild rice and 95% long-grain rice and the deluxe rice mixture is 10% wild rice and 90% long-grain rice.The company has just received a shipment of 120 pounds of wild rice and 1,500pounds of long-grain rice. How many pounds of each type of rice mixture shouldbe produced to use all the rice in this shipment?

Airspeed

An airplane makes the 2,400-mile trip from Washington, D.C., to San Fran-cisco in 7.5 hours and makes the return trip in 6 hours. Assuming that theplane travels at a constant airspeed and that the wind blows at a constant ratefrom west to east, find the plane’s airspeed and the wind rate.

S o l u t i o n Let x represent the airspeed of the plane and let y represent the rate at which thewind is blowing (both in miles per hour). The ground speed of the plane is deter-mined by combining these two rates; that is,

x 2 y 5 Ground speed flying east to west (headwind)

x 1 y 5 Ground speed flying west to east (tailwind)

Applying the familiar formula D 5 RT to each leg of the trip leads to the fol-lowing system of equations:

2,400 5 7.5(x 2 y) From Washington to San Francisco

2,400 5 6(x 1 y) From San Francisco to Washington

After simplification, we have

x 2 y 5 320

x 1 y 5 400

Solve using substitution:

Solve first equation for x.

Substitute in second equation.

Wind rate y 5 40 mph

2y 5 80

y 1 320 1 y 5 400

x 5 y 1 320

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6

FIGURE 3


Airspeed

C h e c k

A boat takes 8 hours to travel 80 miles upstream and 5 hours to return to its start-ing point. Find the speed of the boat in still water and the speed of the current.

Supply and Demand

The quantity of a product that people are willing to buy during some periodof time depends on its price. Generally, the higher the price, the less thedemand; the lower the price, the greater the demand. Similarly, the quantityof a product that a supplier is willing to sell during some period of time alsodepends on the price. Generally, a supplier will be willing to supply more ofa product at higher prices and less of a product at lower prices. This exampleuses linear models to analyze the relationship between supply and demand.

S o l u t i o n Suppose we are interested in analyzing the sale of cherries each day in a partic-ular city. An analyst arrives at the following price–demand and price–supplyequations:

p 5 20.3q 1 5 Demand equation (consumer)

p 5 0.06q 1 0.68 Supply equation (supplier)

where q represents the quantity in thousands of pounds and p represents the pricein dollars. The graphs of these equations are shown in Figure 4(a), where we havesubstituted x for q.

y1 5 20.3x 1 5 y3 5 1.70 y4 5 1.10y2 5 0.06x 1 0.68 Supply exceeds demand Demand exceeds supply

(a) (b) (c)FIGURE 4

Suppose that cherries are selling for $1.70 per pound. Using a built-in inter-section routine (details omitted), we find that the horizontal line p 5 1.70 inter-sects the demand equation at q 5 11 and the supply equation at q 5 17 [see

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7

2,400 ⁄ 2,400 2,400 ⁄ 2,400

2,400 ‚ 6(360 1 40) 2,400 ‚ 7.5(360 2 40)

2,400 5 6(x 1 y) 2,400 5 7.5(x 2 y)

x 5 360 mph

x 5 40 1 320


Fig. 4(b)]. Thus, at a price of $1.70 per pound, consumers will purchase 11,000pounds of cherries and suppliers are willing to supply 17,000 pounds. The sup-ply exceeds the demand at this price, and the price will come down. Now sup-pose that the price drops to $1.10 per pound [Fig. 4(c)]. Proceeding as before(details omitted), we find that at this price consumers will purchase 13,000 poundsof cherries, but suppliers will supply only 7,000 pounds. Thus, at $1.10 per poundthe demand exceeds the supply and the price will go up. At what price will cher-ries stabilize for the day? That is, at what price will supply equal demand? Thisprice, if it exists, is called the equilibrium price, the quantity sold at that priceis called the equilibrium quantity, and the point of intersection of the supply anddemand equations is called the equilibrium point. Using a built-in intersectionroutine (Fig. 5), we see that the equilibrium quantity is 12,000 pounds and theequilibrium price is $1.40.

The price–demand and price–supply equations for strawberries in a certain cityare

p 5 20.2q 1 4 Demand equation

p 5 0.04q 1 1.84 Supply equation

where q represents the quantity in thousands of pounds and p represents the pricein dollars.

Find the equilibrium quantity and the equilibrium price.

A n s w e r s t o M a t c h e d P r o b l e m s

1. x 5 1, y 5 22Check:

2. (A) (3, 2) or x 5 3 and y 5 2 (B) No solutions (C) Infinite number of solutions3. (21.92, 4.23) or x 5 21.92 and y 5 4.23 4. x 5 1, y 5 22 5. x 5 2, y 5 236. 840 pounds of regular mix, 780 pounds of deluxe mix7. Boat: 13 mph, current: 3 mph8. Equilibrium quantity 5 9 thousand pounds; Equilibrium price 5 $2.20 per pound

23 ⁄ 23

1 1 2(22) ‚ 23

x 1 2y 5 23 3 ⁄ 3

1 2 (22) ‚ 3

x 2 y 5 3


8

FIGURE 5

Equilibrium point


EXERCISE 8-1

A

Match each system in Problems 1–4 with one of the followinggraphs, and use the graph to solve the system.

(a) (b)

(c) (d)

1. 2x 2 4y 5 8 2. x 1 y 5 3x 2 2y 5 0 x 2 2y 5 0

3. 2x 2 y 5 5 4. 4x 2 2y 5 103x 1 2y 5 23 2x 2 y 5 5

Solve Problems 5–10 by graphing.

5. x 1 y 5 7 6. x 2 y 5 2x 2 y 5 3 x 1 y 5 4

7. 3x 2 2y 5 12 8. 3x 2 y 5 27x 1 2y 5 8 x 1 2y 5 10

9. 3u 1 5v 5 15 10. m 1 2n 5 46u 1 10v 5 230 2m 1 4n 5 28

Solve Problems 11–16 by substitution.

11. y 5 2x 1 3 12. y 5 x 1 4 13. x 2 y 5 4y 5 3x 2 5 y 5 5x 2 8 x 1 3y 5 12

14. 2x 2 y 5 3 15. 3x 2 y 5 7 16. 2x 1 y 5 6x 1 2y 5 14 2x 1 3y 5 1 x 2 y 5 23

B

Solve Problems 17–30 by either method. Round anyapproximate values to two decimal places.

17. 4x 1 3y 5 26 18. 9x 2 3y 5 243x 2 11y 5 27 11x 1 2y 5 1

19. 7m 1 12n 5 21 20. 3p 1 8q 5 45m 2 3n 5 7 15p 1 10q 5 210

21. y 5 0.08x 22. y 5 0.07xy 5 100 1 0.04x y 5 80 1 0.05x

23. 0.2u 2 0.5y 5 0.07 24. 0.3s 2 0.6t 5 0.180.8u 2 0.3v 5 0.79 0.5s 2 0.2t 5 0.54

25. 5 2 26. 5 10

5 25 5 6

27. 2x 2 3y 5 25 28. 7x 2 3y 5 203x 1 4y 5 13 5x 1 2y 5 8

29. 3.5x 2 2.4y 5 0.1 30. 5.4x 1 4.2y 5 212.92.6x 2 1.7y 5 20.2 3.7x 1 6.4y 5 24.5

31. In the process of solving a system by substitution, supposeyou encounter a contradiction, such as 0 5 1. How wouldyou describe the solutions to such a system? Illustrateyour ideas with the system

x 2 2y 5 23

22x 1 4y 5 7

32. In the process of solving a system by substitution, supposeyou encounter an identity, such as 0 5 0. How would youdescribe the solutions to such a system? Illustrate yourideas with the system

x 2 2y 5 23

22x 1 4y 5 6

C

In Problems 33 and 34, solve each system for p and q in termsof x and y. Explain how you could check your solution andthen perform the check.

33. x 5 2 1 p 2 2q 34. x 5 21 1 2p 2 qy 5 3 2 p 1 3q y 5 4 2 p 1 q

Problems 35 and 36 refer to the system

ax 1 by 5 h

cx 1 dy 5 k

where x and y are variables and a, b, c, d, h, and k are realconstants.

25 x 1 4

3 y73 x 2 5

4 y

72 x 2 5

6 y25 x 1 3

2 y


35. Solve the system for x and y in terms of the constants a, b,c, d, h, and k. Clearly state any assumptions you mustmake about the constants during the solution process.

36. Discuss the nature of solutions to systems that do not sat-isfy the assumptions you made in Problem 35.

APPLICATIONS

37. Airspeed.It takes a private airplane 8.75 hours to makethe 2,100-mile flight from Atlanta to Los Angeles and 5hours to make the return trip. Assuming that the windblows at a constant rate from Los Angeles to Atlanta, findthe airspeed of the plane and the wind rate.

38. Airspeed.A plane carries enough fuel for 20 hours offlight at an airspeed of 150 miles per hour. How far can itfly into a 30 mph headwind and still have enough fuel toreturn to its starting point? (This distance is called thepoint of no return.)

39. Rate––Time. A crew of eight can row 20 kilometers perhour in still water. The crew rows upstream and then re-turns to its starting point in 15 minutes. If the river is flow-ing at 2 kilometers per hour, how far upstream did thecrew row?

40. Rate––Time. It takes a boat 2 hours to travel 20 milesdown a river and 3 hours to return upstream to its startingpoint. What is the rate of the current in the river?

41. Chemistry.A chemist has two solutions of hydrochloricacid in stock: a 50% solution and an 80% solution. Howmuch of each should be used to obtain 100 milliliters of a68% solution?

42. Business.A jeweler has two bars of gold alloy in stock,one of 12 carats and the other of 18 carats (24-carat gold ispure gold, 12-carat is pure, 18-carat gold is pure, andso on). How many grams of each alloy must be mixed toobtain 10 grams of 14-carat gold?

43. Finance.Suppose you have $12,000 to invest. If part isinvested at 10% and the rest at 15%, how much should beinvested at each rate to yield 12% on the total amountinvested?

44. Finance.An investor has $20,000 to invest. If part is in-vested at 8% and the rest at 12%, how much should beinvested at each rate to yield 11% on the total amountinvested?

45. Production.A supplier for the electronics industry manu-factures keyboards and screens for graphing calculators atplants in Mexico and Taiwan. The hourly production ratesat each plant are given in the table. How many hoursshould each plant be operated to exactly fill an order for4,000 keyboards and screens?

1824

1224

Plant Keyboards Screens

Mexico 40 32

Taiwan 20 32

46. Production.A company produces Italian sausages andbratwursts at plants in Green Bay and Sheboygan. Thehourly production rates at each plant are given in the table.How many hours should each plant be operated to exactlyfill an order for 62,250 Italian sausages and 76,500bratwursts?

ItalianPlant Sausage Bratwurst

Green Bay 800 800

Sheboygan 500 1,000

47. Nutrition. Animals in an experiment are to be kept on astrict diet. Each animal is to receive, among other things,20 grams of protein and 6 grams of fat. The laboratorytechnician is able to purchase two food mixes of the fol-lowing compositions: Mix A has 10% protein and 6% fat;mix B has 20% protein and 2% fat. How many grams ofeach mix should be used to obtain the right diet for a sin-gle animal?

48. Nutrition. A fruit grower can use two types of fertilizer inan orange grove, brand A and brand B. Each bag of brandA contains 8 pounds of nitrogen and 4 pounds of phos-phoric acid. Each bag of brand B contains 7 pounds of ni-trogen and 7 pounds of phosphoric acid. Tests indicate thatthe grove needs 720 pounds of nitrogen and 500 pounds ofphosphoric acid. How many bags of each brand should beused to provide the required amounts of nitrogen andphosphoric acid?

49. Supply and Demand.Suppose the supply and demandequations for printed T-shirts in a resort town for a particu-lar week are

p 5 0.007q 1 3 Supply equation


where p is the price in dollars and q is the quantity.

(A) Find the supply and the demand (to the nearest unit) ifT-shirts are priced at $4 each. Discuss the stability ofthe T-shirt market at this price level.

(B) Find the supply and the demand (to the nearest unit) ifT-shirts are priced at $8 each. Discuss the stability ofthe T-shirt market at this price level.

(C) Find the equilibrium price and quantity.

(D) Graph the two equations in the same coordinatesystem and identify the equilibrium point, supplycurve, and demand curve.

8-2 Systems of Linear Equations and Augmented Matrices 595

50. Supply and Demand.Suppose the supply and demand forprinted baseball caps in a resort town for a particular weekare

p 5 20.006q 1 2 Supply equation


where p is the price in dollars and q is the quantity inhundreds.

(A) Find the supply and the demand (to the nearest unit) ifbaseball caps are priced at $4 each. Discuss thestability of the baseball cap market at this price level.

(B) Find the supply and the demand (to the nearest unit) ifbaseball caps are priced at $8 each. Discuss thestability of the baseball cap market at this price level.


(D) Graph the two equations in the same coordinatesystem and identify the equilibrium point, supplycurve, and demand curve.

★ 51. Supply and Demand.At $0.60 per bushel, the daily sup-ply for wheat is 450 bushels and the daily demand is 645bushels. When the price is raised to $0.90 per bushel, thedaily supply increases to 750 bushels and the daily de-mand decreases to 495 bushels. Assume that the supplyand demand equations are linear.

(A) Find the supply equation. [Hint: Write the supplyequation in the form p 5 aq 1 b and solve for a andb.]

(B) Find the demand equation.


★ 52. Supply and Demand.At $1.40 per bushel, the daily sup-ply for soybeans is 1,075 bushels and the daily demand is580 bushels. When the price falls to $1.20 per bushel, thedaily supply decreases to 575 bushels and the daily de-mand increases to 980 bushels. Assume that the supplyand demand equations are linear.

(A) Find the supply equation. [See the hint in Problem51.]

(B) Find the demand equation.


★ 53. Physics.An object dropped off the top of a tall buildingfalls vertically with constant acceleration. If s is the dis-tance of the object above the ground (in feet) t seconds af-ter its release, then sand t are related by an equation of theform

s 5 a 1 bt2

where a and b are constants. Suppose the object is 180 feetabove the ground 1 second after its release and 132 feetabove the ground 2 seconds after its release.

(A) Find the constants a and b.

(B) How high is the building?

(C) How long does the object fall?

★ 54. Physics.Repeat Problem 53 if the object is 240 feet abovethe ground after 1 second and 192 feet above the groundafter 2 seconds.

★ 55. Earth Science.An earthquake emits a primary wave and asecondary wave. Near the surface of the Earth the primarywave travels at about 5 miles per second and the second-ary wave at about 3 miles per second. From the time lagbetween the two waves arriving at a given receiving sta-tion, it is possible to estimate the distance to the quake.(The epicentercan be located by obtaining distance bear-ings at three or more stations.) Suppose a station measureda time difference of 16 seconds between the arrival of thetwo waves. How long did each wave travel, and how farwas the earthquake from the station?

★ 56. Earth Science.A ship using sound-sensing devices aboveand below water recorded a surface explosion 6 secondssooner by its underwater device than its above-water de-vice. Sound travels in air at about 1,100 feet per secondand in seawater at about 5,000 feet per second.

(A) How long did it take each sound wave to reach theship?

(B) How far was the explosion from the ship?

Section 8-2 Systems of Linear Equations and Augmented Matrices

Elimination by AdditionMatricesSolving Linear Systems Using Augmented Matrices

Most real-world applications of linear systems involve a large number of vari-ables and equations. Computers are usually used to solve these larger systems.Although very effective for systems involving two variables, the graphing and

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