Solve the linear systems in Exercises 13 through 17. You may use technology

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

In Exercises 1 through 12, find all solutions of the equations with paper and pencil using Gauss-Jordan elimination. Show all your work. Solve the system in Exercise 8 for the variables x\, xi, *3 , *4 , and x$.

Solve the linear systems in Exercises 13 through 17. You may use technology.

Solve the linear systems in Exercises 13 through 17. You may use technology.

Solve the linear systems in Exercises 13 through 17. You may use technology.

Solve the linear systems in Exercises 13 through 17. You may use technology.

Solve the linear systems in Exercises 13 through 17. You may use technology.

Determine which of the matrices below are in reduced row-echelon form: a. 1 2 0 2 0 0 0 1 3 0 0 0 1 4 0 0 0 0 0 1 1 2 0 3 0 0 0 0 0 0 1 2 0 1 2 0 3" b. 0 0 0 1 4 0 0 0 0 0 d. [0 1 2 3 4]

Find all 4 x 1 matrices in reduced row-echelon form.

We say that two n x m matrices in reduced row-echelon form are of the same type if they contain the same number of leading l s in the same positions. For example, ' 2 O ' CD 3 0" 0 0 and 0 0 are of the same type. How many types of 2 x 2 matrices in reduced row-echelon form are there?

How many types of 3 x 2 matrices in reduced rowechelon form are there? (See Exercise 20.)

How many types of 2 x 3 matrices in reduced rowechelon form are there? (See Exercise 20.)

Suppose you apply Gauss-Jordan elimination to a matrix. Explain how you can be sure that the resulting matrix is in reduced row-echelon form.

Suppose matrix A is transformed into matrix B by means of an elementary row operation. Is there an elementary row operation that transforms B into A1 Explain.

Suppose matrix A is transformed into matrix B by a sequence of elementary row operations. Is there a sequence of elementary row operations that transforms B into A? Explain your answer. (See Exercise 24.)

Consider an n x m matrix A. Can you transform rref( A) into A by a sequence of elementary row operations? (See Exercise 25.)

Is there a sequence of elementary row operations that transforms "1 2 3 'l 0 0" 4 5 6 into 0 1 0 7 8 9 0 0 0 Explain.

Suppose you subtract a multiple of an equation in a system from another equation in the system. Explain why the two systems (before and after this operation) have the same solutions.

Balancing a chemical reaction. Consider the chemical reaction a N02 + b H20 -* c HN02 4- d HNO3, where a, b, c\ and d arc unknown positive integers. The reaction must be balanced; that is, the number of atoms of each element must be the same before and after the reaction. For example, because the number of oxygen atoms must remain the same, 2a 4* b = 2c 3d. While there are many possible values for a , b, t\ and d that balance the reaction, it is customary to use the smallest possible positive integers. Balance this reaction.

Find the polynomial of degree 3 [a polynomial of the form f(t) = a + bt + ct2 + d t3] whose graph goes through the points (0, 1), (1, 0), ( 1.0), and (2, 15). Sketch the graph of this cubic.

Find the polynomial of degree 4 whose graph goes through the points (1, 1), (2, 1), (3, 59), (1,5), and (2. 29). Graph this polynomial.

Cubic splines. Suppose you are in charge of the design of a roller coaster ride. This simple ride will not make any left or right turns; that is, the track lies in a vertical plane. The accompanying figure shows the ride as viewed from the side. The points (a, , /?,) are given to you, and your job is to connect the dots in a reasonably smooth way. Let aj+ \ > ai.One method often employed in such design problems is the technique of cubic splines. We choose ft (r), a polynomial of degree < 3, to define the shape of the ride between (a, _ i, _ 1) and (a,-,/?/), for / = l,,..,/j. Obviously, it is required that ft (at) = bi and ft (a/_ i) = bj - 1, for i = 1,..., n. To guarantee a smooth ride at the points (ai,bi), we want the first and the second derivatives of ft and //+ 1 to agree at these points: = //+](,) fliat) = f!'+x(a;). and for / = 1,. ,n - 1. Explain the practical significance of these conditions. Explain why, for the convenience of the riders, it is also required that f\ (00) = /> i.) = 0. Show that satisfying all these conditions amounts to solving a system of linear equations. How many variables are in this system? How many equations? (Note: It can be shown that this system has a unique solution.)

Find the polynomial f(t) of degree 3 such that/(l) = 1, /(2) = 5, /'(l) = 2, and /'(2) = 9, where f( t) is the derivative of f(t). Graph this polynomial.

The dot product of two vectors *1" ">1" x2 y2 and v = _yn_ in R" is defined by x - y = x\y\ + *2^2 + ' + XnVn Note that the dot product of two vectors is a scalar. We say that the vectors x and y are perpendicular if Jc -y = 0 . Find all vectors in R3 perpendicular to Draw a sketch.

Find all vectors in ] vectors

Find all solutions x\, *2, *3 of the equation b = X ] V \ + X 2 V 2 + * 3 ? 3 > where

For some background on this exercise, see Exercise 1.1.20. Consider an economy with three industries, Ij, I2, I3. What outputs jci , *2, *3 should they produce to satisfy both consumer demand and interindustry demand? The demands put on the three industries are shown in the accompanying figure.

If we consider more than three industries in an input- output model, it is cumbersome to represent all the demands in a diagram as in Exercise 37. Suppose we have the industries 11, 12,..., Iw, with outputs jcj , X2, The output vector is The consumer demand vector is V b2 b = bn where bi is the consumer demand on industry I, . The demand vector for industry I j is a \j a2j unj where aij is the demand industry Iy puts on industry 1/, for each $1 of output industry I j produces. For example, 032 = 0.5 means that industry I2 needs 50tfworth of products from industry I3 for each $1 worth of goods I2 produces. The coefficient an need not be 0: Producing a product may require goods or services from the same industry. a. Find the four demand vectors for the economy in Exercise 37. b. What is the meaning in economic terms of xj vj ? c. What is the meaning in economic terms of * 1?1 + * 2?2 H-------h xnvn + b? d. What is the meaning in economic terms of the equation *1^1 + *202 H--------\-xnvn + b = X?

Consider the economy of Israel in 1958.11 The three industries considered here are I[ : agriculture, 12 : manufacturing, 13 : energy. Outputs and demands are measured in millions of Israeli pounds, the currency of Israel at that time. We are told that "13.2 "0.293 b = 17.6 01 = 0.014 - L8_ _0.044_ '0 '0 v2 = 0.207 * h = 0.017 0.01 0.216 a* Why do the first components of v2 and J3 equal 0? b. Find the outputs x\,x2yx^ required to satisfy demand.

Consider some particles in the plane with position vectors rj, ?2,..., rn and masses m j, m2.......mn. The position vector of the center of mass of this system is 1 = T7^wlrl + m2'*2 + -----\~mnrn), M where M = m\ + m2 H-----+ mn. Consider the triangular plate shown in the accompanying sketch. How must a total mass of 1 kg be distributed among the three vertices of the plate so that r 2l the plate can be supported at the point ^ ; that is, r cm ? Assume that the mass of the plate itself is negligible.

The momentum P of a system of n particles in space with masses mi, m2,..., mn and velocities 0i, v2,..., vn is defined as P = m\d\ + m2v2 H--------f-mnvn. Now consider two elementary particles with velocities The momentum P of a system of n particles in space with masses mi, m2,..., mn and velocities 0i, v2,..., vn is defined as P = m\d\ + m2v2 H--------f-mnvn. Now consider two elementary particles with velocities

The accompanying sketch represents a maze of one way streets in a city in the United States. The traffic volume through certain blocks during an hour has been measured. Suppose that the vehicles leaving the area during this hour were exactly the same as those entering it. What can you say about the traffic volume at the four locations indicated by a question mark? Can you figure out exactly how much traffic there was on each block? If not, describe one possible scenario. For each of the four locations, find the highest and the lowest possible traffic volume.

Let S(t) be the length of the fth day of the year 2009 in Mumbai (formerly known as Bombay), India (measured in hours, from sunrise to sunset). We are given the following values of S(t): t S(t) 47 11.5 74 12 273 12 For example, 5(47) = 11.5 means that the time from sunrise to sunset on February 16 is 11 hours and 30 minutes. For locations close to the equator, the function S(t) is well approximated by a trigonometric function of the form S(, ) = + * c o s ^ + c i n ^ ) . (The period is 365 days, or 1 year.) Find this approximation for Mumbai, and graph your solution. According to this model, how long is the longest day of the year in Mumbai?

Kyle is getting some flowers for Olivia, his Valentine. Being of a precise analytical mind, he plans to spend exactly $24 on a bunch of exactly two dozen flowers. At the flower market they have lilies ($3 each), roses ($2 each), and daisies ($0.50 each). Kyle knows that Olivia loves lilies; what is he to do?

Consider the equations * + 2y + 3z = 4 x + ky + 4 2 = 6 , x + 2y + (k + 2)z = 6 where k is an arbitrary constant. a. For which values of the constant k does this system have a unique solution? b. When is there no solution? c. When are there infinitely many solutions?

Consider the equations y + 2kz = 0 x 4- 2y 4- 6z = 2 , kx + 2z = 1 where k is an arbitrary constant. a. For which values of the constant k does this system have a unique solution? b. When is there no solution? c. When are there infinitely many solutions?

a. Find all solutions jq, x2, *3, *4 of the system x2 = j(x i + * 3), *3 = \(X2+X4). b. In part (a), is there a solution with x\ = 1 and X4 = 13?

For an arbitrary positive integer n > 3, find all solutions *1, x2, *3,..., xn of the simultaneous equations x2 = 5(*l+*3),-*3 = 5 (JC2+X4),...,Xn_| = 3(*_2 +*). Note that we are asked to solve the simultaneous equations

Consider the system 2x + y = C 3 y + z = C , jc + 4z = C where C is a constant. Find the smallest positive integer C such that jc, v , and z are all integers.

Find all the polynomials f(t) of degree < 3 such that ' /(0) = 3, /(l) = 2, /(2) = 0, and fQ2 f(t)dt = 4. (If you have studied Simpsons rule in calculus, explain the result.)

Exercises 51 through 60 are concerned with conics. A conic is a curve in M2 that can be described by an equation of the form fix, y) = cj + c2x + c3y + c4x2 + csxy + C6y2 = o, where at least one of the coefficients c, is nonzero. Examples are circles, ellipses, hyperbolas, and parabolas. If k is any nonzero constant, then the equations f(x, y) = 0 and kf(x, y) = 0 describe the same conic. For example, the equation -4 + x2 + y2 = 0 and 12 + 3x2 + 3y2 = 0 both describe the circle of radius 2 centered at the origin. In Exercises 51 through 60, find all the conics through the given points, and draw a rough sketch of your solution curve(s).(0,0), (1,0), (2,0), (0,1), and (0,2).

Exercises 51 through 60 are concerned with conics. A conic is a curve in M2 that can be described by an equation of the form fix, y) = cj + c2x + c3y + c4x2 + csxy + C6y2 = o, where at least one of the coefficients c, is nonzero. Examples are circles, ellipses, hyperbolas, and parabolas. If k is any nonzero constant, then the equations f(x, y) = 0 and kf(x, y) = 0 describe the same conic. For example, the equation -4 + x2 + y2 = 0 and 12 + 3x2 + 3y2 = 0 both describe the circle of radius 2 centered at the origin. In Exercises 51 through 60, find all the conics through the given points, and draw a rough sketch of your solution curve(s). (0,0), (2,0), (0,2), (2,2), and (1,3).

Exercises 51 through 60 are concerned with conics. A conic is a curve in M2 that can be described by an equation of the form fix, y) = cj + c2x + c3y + c4x2 + csxy + C6y2 = o, where at least one of the coefficients c, is nonzero. Examples are circles, ellipses, hyperbolas, and parabolas. If k is any nonzero constant, then the equations f(x, y) = 0 and kf(x, y) = 0 describe the same conic. For example, the equation -4 + x2 + y2 = 0 and 12 + 3x2 + 3y2 = 0 both describe the circle of radius 2 centered at the origin. In Exercises 51 through 60, find all the conics through the given points, and draw a rough sketch of your solution curve(s). (0,0), (1,0), (2,0), (3,0), and (1,1).

Exercises 51 through 60 are concerned with conics. A conic is a curve in M2 that can be described by an equation of the form fix, y) = cj + c2x + c3y + c4x2 + csxy + C6y2 = o, where at least one of the coefficients c, is nonzero. Examples are circles, ellipses, hyperbolas, and parabolas. If k is any nonzero constant, then the equations f(x, y) = 0 and kf(x, y) = 0 describe the same conic. For example, the equation -4 + x2 + y2 = 0 and 12 + 3x2 + 3y2 = 0 both describe the circle of radius 2 centered at the origin. In Exercises 51 through 60, find all the conics through the given points, and draw a rough sketch of your solution curve(s). (0,0), (1,1), (2,2), (3, 3), and (1.0).

Exercises 51 through 60 are concerned with conics. A conic is a curve in M2 that can be described by an equation of the form fix, y) = cj + c2x + c3y + c4x2 + csxy + C6y2 = o, where at least one of the coefficients c, is nonzero. Examples are circles, ellipses, hyperbolas, and parabolas. If k is any nonzero constant, then the equations f(x, y) = 0 and kf(x, y) = 0 describe the same conic. For example, the equation -4 + x2 + y2 = 0 and 12 + 3x2 + 3y2 = 0 both describe the circle of radius 2 centered at the origin. In Exercises 51 through 60, find all the conics through the given points, and draw a rough sketch of your solution curve(s). (0,0), (1,0), (0,1), and (1,1).

Exercises 51 through 60 are concerned with conics. A conic is a curve in M2 that can be described by an equation of the form fix, y) = cj + c2x + c3y + c4x2 + csxy + C6y2 = o, where at least one of the coefficients c, is nonzero. Examples are circles, ellipses, hyperbolas, and parabolas. If k is any nonzero constant, then the equations f(x, y) = 0 and kf(x, y) = 0 describe the same conic. For example, the equation -4 + x2 + y2 = 0 and 12 + 3x2 + 3y2 = 0 both describe the circle of radius 2 centered at the origin. In Exercises 51 through 60, find all the conics through the given points, and draw a rough sketch of your solution curve(s). (0,0), (1,0), (0,1), and (1,-1).

Exercises 51 through 60 are concerned with conics. A conic is a curve in M2 that can be described by an equation of the form fix, y) = cj + c2x + c3y + c4x2 + csxy + C6y2 = o, where at least one of the coefficients c, is nonzero. Examples are circles, ellipses, hyperbolas, and parabolas. If k is any nonzero constant, then the equations f(x, y) = 0 and kf(x, y) = 0 describe the same conic. For example, the equation -4 + x2 + y2 = 0 and 12 + 3x2 + 3y2 = 0 both describe the circle of radius 2 centered at the origin. In Exercises 51 through 60, find all the conics through the given points, and draw a rough sketch of your solution curve(s). (5,0), (1,2), (2, 1), (8,1), and (2,9).

Exercises 51 through 60 are concerned with conics. A conic is a curve in M2 that can be described by an equation of the form fix, y) = cj + c2x + c3y + c4x2 + csxy + C6y2 = o, where at least one of the coefficients c, is nonzero. Examples are circles, ellipses, hyperbolas, and parabolas. If k is any nonzero constant, then the equations f(x, y) = 0 and kf(x, y) = 0 describe the same conic. For example, the equation -4 + x2 + y2 = 0 and 12 + 3x2 + 3y2 = 0 both describe the circle of radius 2 centered at the origin. In Exercises 51 through 60, find all the conics through the given points, and draw a rough sketch of your solution curve(s). (1.0), (2,0), (2,2), (5, 2). and (5, 6).

Exercises 51 through 60 are concerned with conics. A conic is a curve in M2 that can be described by an equation of the form fix, y) = cj + c2x + c3y + c4x2 + csxy + C6y2 = o, where at least one of the coefficients c, is nonzero. Examples are circles, ellipses, hyperbolas, and parabolas. If k is any nonzero constant, then the equations f(x, y) = 0 and kf(x, y) = 0 describe the same conic. For example, the equation -4 + x2 + y2 = 0 and 12 + 3x2 + 3y2 = 0 both describe the circle of radius 2 centered at the origin. In Exercises 51 through 60, find all the conics through the given points, and draw a rough sketch of your solution curve(s). (0,0), (1.0), (2,0), (0, 1), (0,2), and (1, 1).

Exercises 51 through 60 are concerned with conics. A conic is a curve in M2 that can be described by an equation of the form fix, y) = cj + c2x + c3y + c4x2 + csxy + C6y2 = o, where at least one of the coefficients c, is nonzero. Examples are circles, ellipses, hyperbolas, and parabolas. If k is any nonzero constant, then the equations f(x, y) = 0 and kf(x, y) = 0 describe the same conic. For example, the equation -4 + x2 + y2 = 0 and 12 + 3x2 + 3y2 = 0 both describe the circle of radius 2 centered at the origin. In Exercises 51 through 60, find all the conics through the given points, and draw a rough sketch of your solution curve(s). (0,0), (2,0), (0,2), (2, 2), (1,3), and (4, 1).

Students are buying books for the new semester. Eddie buys the environmental statistics book and the set theory book for $178. Leah, who is buying books for herself and her friend, spends $319 on two environmental statistics books, one set theory book, and one educational psychology book. Mehmet buys the educational psychology book and the set theory book for $147 in total. How much does each book cost?

Students are buying books for the new semester. Brigitte buys the German grammar book and the German novel, Die Leiden des jungen Wert her, for 64 in total. Claude spends 98 on the linear algebra text and the German grammar book, while Denise buys the linear algebra text and Werther, for 76. How much does each of the three books cost?

At the beginning of a political science class at a large university, the students were asked which term, liberal or conservative, best described their political views. They were asked the same question at the end of the course, to see what effect the class discussions had on their views. Of those that characterized themselves as liberal initially, 30% held conservative views at the end. Of those who were conservative initially, 40% moved to the liberal camp. It turned out that there were just as many students with conservative views at the end as there had been liberal students at the beginning. Out of the 260 students in the class, how many held liberal and conservative views at the beginning of the course and at the end? (No students joined or dropped the class between the surveys, and they all participated in both surveys.)

At the beginning of a semester, 55 students have signed up for Linear Algebra; the course is offered in two sections that are taught at different times. Because of scheduling conflicts and personal preferences, 20% of the students in Section A switch to Section B in the first few weeks of class, while 30% of the students in Section B switch to A, resulting in a net loss of 4 students for Section B. How large were the two sections at the beginning of the semester? No students dropped Linear Algebra (why would they?) or joined the course late.

Five cows and two sheep together cost ten liang12 of silver. Two cows and five sheep together cost eight liang of silver. What is the cost of a cow and a sheep, respectively? (Nine Chapters,13 Chapter 8, Problem 7)

If you sell two cows and five sheep and you buy 13 pigs, you gain 1,000 coins. If you sell three cows and three pigs and buy nine sheep, you break even. If you sell six sheep and eight pigs and you buy five cows, you lose 600 coins. What is the price of a cow, a sheep, and a pig, respectively? (Nine Chapters, Chapter 8, Problem 8)

You place five sparrows on one of the pans of a balance and six swallows on the other pan; it turns out that the sparrows are heavier. But if you exchange one sparrow and one swallow, the weights are exactly balanced. All the birds together weigh 1 jin. What is the weight of a sparrow and a swallow, respectively? [Give the answer in liang, with 1 jin = 16 liang.] (Nine Chapters, Chapter 8, Problem 9)

Consider the task of pulling a weight of 40 dan14 up a hill; we have one military horse, two ordinary horses, and three weak horses at our disposal to get the job done. It turns out that the military horse and one of the ordinary horses, pulling together, are barely able to pull the weight (but they could not pull any more). Likewise, the two ordinary horses together with one weak horse are just able to do the job, as are the three weak horses together with the military horse. How much weight can each of the horses pull alone? (Nine Chapters, Chapter 8, Problem 12)

Five households share a deep well for their water supply. Each household owns a few ropes of a certain length, which varies only from household to household. The five households, A, B, C, D, and E, own 2, 3, 4, 5, and 6 ropes, respectively. Even when tying all their ropes together, none of the households alone are able to reach the water, but As two ropes together with one of B\s ropes just reach the water. Likewise, Bs three ropes with one of Cs ropes, Cs four ropes with one of Ds ropes, Ds five ropes with one of Es ropes, and Es six ropes with one of As ropes all just reach the water. How long are the ropes of the various households, and how deep is the well? Commentary: As stated, this problem leads to a system of 5 linear equations in 6 variables; with the given information, we are unable to determine the depth of the well. The Nine Chapters gives one particular solution, where the depth of the well is 7 zhang,'5 2 chi, 1 cun, or 721 cun (since 1 zhang = 10 chi and 1 chi = 10 cun). Using this particular value for the depth of the well, find the lengths of the various ropes.

A rooster is worth five coins, a hen three coins, and 3 chicks one coin. With 100 coins we buy 100 of them. How many roosters, hens, and chicks can we buy? (From the Mathematical Manual by Zhang Qiujian, Chapter 3, Problem 38; 5th century a.d.) Commentary: This famous Hundred Fowl Problem has reappeared in countless variations in Indian, Arabic, and European texts (see Exercises 71 through 74); it has remained popular to this day (see Exercise 44 of this section).

Pigeons are sold at the rate of 5 for 3 panas, sarasabirds at the rate of 7 for 5 panas, swans at the rate of 9 for 7 panas, and peacocks at the rate of 3 for 9 panas. A man was told to bring 100 birds for 100 panas for the amusement of the Kings son. What does he pay for each of the various kinds of birds that he buys? (From the Ganita-Sara-Sangraha by Mahavira, India; 9th century A.D.) Find one solution to this problem.

A duck costs four coins, five sparrows cost one coin, and a rooster costs one coin. Somebody buys 100 birds for 100 coins. How many birds of each kind can he buy? (From the Key to Arithmetic by Al-Kashi; 15th century)

A certain person buys sheep, goats, and hogs, to the number of 100, for 100 crowns; the sheep cost him ^ a crown a-piece; the goats, 1 ^ crown; and the hogs 3^ crowns. How many had he of each? (From the Elements of Algebra by Leonhard Euler, 1770)

A gentleman has a household of 100 persons and orders that they be given 100 measures of grain. He directs that each man should receive three measures, each woman two measures, and each child half a measure. How many men, women, and children are there in this household? We are told that there is at least one man, one woman, and one child. (From the Problems for Quickening a Young Mind by Alcuin [c. 732-804], the Abbot of St. Martins at Tours. Alcuin was a friend and tutor to Charlemagne and his family at Aachen.)

A father, when dying, gave to his sons 30 barrels, of which 10 were full of wine, 10 were half full, and the last 10 were empty. Divide the wine and flasks so that there will be equal division among the three sons of both wine and barrels. Find all the solutions of this problem. (From Alcuin)

Make me a crown weighing 60 minae, mixing gold, bronze, tin, and wrought iron. Let the gold and bronze together form two-thirds, the gold and tin together three- fourths, and the gold and iron three-fifths. Tell me how much gold, tin, bronze, and iron you must put in (From the Greek Anthology by Metrodorus, 6th century A.D.)

Three merchants find a purse lying in the road. One merchant says If I keep the purse, 1 shall have twice as much money as the two of you together Give me the purse and I shall have three times as much as the two of you together said the second merchant. The third merchant said I shall be much better off than either of you if I keep the purse, I shall have five times as much as the two of you together. If there are 60 coins (of equal value) in the purse, how much money does each merchant have? (From Mahavira)

3 cows graze 1 field bare in 2 days, 7 cows graze 4 fields bare in 4 days, and 3 cows graze 2 fields bare in 5 days. It is assumed that each field initially provides the same amount, x, of grass; that the daily growth, y, of the fields remains constant; and that all the cows eat the same amount, z, each day. (Quantities *, >\ and z are measured by weight.) Find all the solutions of this problem. (This is a special case of a problem discussed by Isaac Newton in his Arithmetica Universalis, 1707.)