Subtracting the members of equation (2) from the corresponding members of (1), we obtain Or, x2- y2+2yz - 2 zx = 0. (6) (7) The system composed of equations (5) and (7), and any one of the given equations, such as (1), is equivalent to the given system of equations. [x + y + z + 2][x + y + z − 2] = 0, (5) (x − y)[x + y −2%]=0, x2 + 2 yz (7) II. = 1. Equivalent Equating the factors of the first members of equations (5) and (7) separately to zero, and applying the method of § 16, we may separate the derived System II. into a group of four derived systems, which taken together are equivalent to System II. Solving these systems separately, the solutions of the given system of equations may be obtained. From the first equation we may obtain the value of x, expressed in terms of y, as follows: Substituting this value for x in equation (3), we obtain an equation containing y and z the members of which may be combined with those of equation (2) to obtain the values of y and z. The values of x may be obtained by substituting in equation (4). Ex. 6. Solve the system of equations If, from the squares of the members of equation (1) we subtract the product of the corresponding members of equations (2) and (3), we shall obtain equation (4). Equations (5) and (6) may be obtained in a similar way. x [x3 + y3 + z3 — 3 xyz] = a2 — bc. (4) (6) From equations (4) and (6) we obtain by division the value of the fraction x/z. Similarly, from equations (5) and (6) we obtain the value of the fraction y/z. From the equations thus obtained we may find the expressed values of x and y in terms of z. Substituting for x and y in equation (1) their expressed values thus found, we obtain the value of z in terms of the known numbers a, b, and c, in the form of a fraction having an irrational denominator, %= +(c2-ab) √ a3 + b3 + c3 - 3 abc (7) Either by substituting this value for z in the remaining equations, which may then be solved for x and y, or by repeating the process above with different pairs of equations, we obtain expressions of the same type as (7) for x and y. EXERCISE XXV. 8 Find sets of values which satisfy each of the following systems of equations: Solve the following problems employing conditional equations containing two or more unknown quantities: 1. Find two numbers the sum of which is 40, and the product of which is 256. 2. Find two numbers the difference of which is 15, and the sum of the squares of which is 293. 3. Find two numbers the difference of which is 6, and the product of which is 247. 4. Find two numbers the sum of which is 40, and the product of which is 391. 5. Find two numbers the product of which is 96, and the sum of the squares of which is 208. 6. Find two numbers the difference of the squares of which is 112, and the square of the difference of which is 64. 7. Find two numbers the sum of which is 10, and the sum of the cubes of which is 370. 8. The product of two numbers is 64, and the quotient obtained by dividing the greater number by the less is 4. Find the numbers. 9. If the sum of two numbers is divided by the less number, the quotient is 4; the product of the numbers is 27. Find the numbers. 10. Find two numbers the sum of which is 20, such that the sum of the quotients obtained by dividing each number by the other is 17/4. 11. Find two numbers such that, if each be increased by 1, the product is 124, and the product obtained by multiplying the first number by a number less by one than the second number is 60. 12. Find two numbers the sum of which is twice their difference, and the difference of the squares of which is 200. 13. The product of two numbers is 12, and the sum of their squares is five times the sum of the numbers. Find the numbers. 14. Find two numbers, of which the sum is 14, which are such that the product of the first and the reciprocal of the second, increased by the product of the second and the reciprocal of the first, is 25/12. 15. The sum of two numbers is 30, and the sum of the quotients resulting from dividing each number by the other is 82/9. Find the numbers. 16. The first of two numbers is ten times the reciprocal of the second, and the sum of the second number and ten times the reciprocal of the first is equal to the square of the second number. Find the numbers. 17. Find two fractions such that the sum of the first fraction and the reciprocal of the second is equal to 2, and the sum of the second fraction and the reciprocal of the first is 8/3. 18. Find two numbers the sum of which is 36, and half the product of which is equal to the cube of the less number. 19. Find a fraction the value of which is 3/4, and the product of the numerator and denominator of which is 48. 20. If a certain two-figure number, the sum of the figures of which is 12, be multiplied by the units' figure, the product is 375. What is the number? 21. A number expressed by two figures is equal to four times the sum of the figures. The number formed by writing the figures in reversed order exceeds three times the product of the figures by the square of the figure in tens' place of the given number. Find the number. 22. If it requires 240 rods of fence to enclose a rectangular field of 20 acres, what are the dimensions of the field? 23. A rectangular field contains 30 acres. By increasing its length by 40 rods and diminishing its width by 4 rods, the area is increased by 6 What are its dimensions? acres. 24. The length of the fence around a rectangular field is 274 yards, and the distance measured diagonally from corner to corner is 97 yards. What is the area? 25. Thirty-two yards of the fence about a rectangular field which is 184 yards long and 76 yards wide are destroyed. What must be the dimensions of a rectangular field in order that the length of fence remaining shall enclose the same area as before? 26. A property owner wishes to use the material from a stone wall enclosing a field, which has the form of a rectangle 80 rods long and 60 rods wide, to build another wall greater by 16 rods which shall enclose a second tract of land which has the form of a rectangle having the same area as the first. Find the dimensions of the second tract of land. 27. In widening a street, a strip of land 6 feet in width was removed from the entire frontage of a tract containing 28,800 square feet. By increasing the frontage of the reduced lot by 8 feet, the entire area became the same as before. Find the original dimensions of the land. 28. It is observed that, if a guy rope which is attached to a stake 7 feet |