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Ans. x=10, y=18. Ex. 11. My shoemaker sends me a bill of $9 for 1 pair of boots and 2 pair of shoes. Some months afterward he sends me a bill of $16 for 2 pair of boots and 3 pair of shoes. What are the boots and shoes a pair ?
Ans. The boots are $5 and the shoes $2. Ex. 12. A gentleman employs 5 men and 4 boys to labor one day, and pays them $7; the next day he hires, at the same wages, 8 men and 6 boys, and pays them $11. What are the daily wages of each? Ans. The men have one dollar and the boys 50
cents. Lx. 13. A vintner sold at one time 15 bottles of port and 10 bottles of sherry, and for the whole received $30. At another time he sold 12 bottles of port and 18 bottles of sherry, at the same prices as before, and for the whole received $39. What was the price of a bottle of each sort of wine ?
Ans. The port was $1 and the sherry $1.50. Ex. 14. A gentleman has two horses and one chaise. The first horse is worth $120. If the first horse be harnessed to the chaise, they will together be worth twice as much as the second horse. But if the second be harnessed, the horse and chaise will be worth three times the value of the first. What is the value of the second horse and of the chaise ? Ans. The second horse is worth $160 and the chaise
$200. Ex. 15. A farmer bought a quantity of rye and
wheat for $18, the rye at 75 cents and the wheat at $1 per bushel. He afterward sold } of his rye and I of his wheat, at the same rate, for $5. How many bushels were there of each?
Ans. 8 bushels of rye and 12 bushels of wheat. (114.) The same example may be solved by either of the preceding methods, and each method has its advantages in particular cases. Generally, however, the first two methods give rise to fractional expressions which occasion inconvenience in practice, while the third method is not liable to this objection. When the coefficient of one of the unknown quantities in one of the equations is equal to unity, this inconvenience does not occur, and the method by substitution may be preferable; the third will, however, commonly be found most convenient. The following examples may be solved by either of these methods.
Ex. 1. Given 5x+4y=68, and 3x+7y=73, to find the values of x and y.
Ex. 5. Given + =26, and -*=8, to find the values of u and y.
Ans. x=70, y=24. Ex. 6. It is required to find two numbers such that if half the first be added to the second, the sum will be 34, and if one third of the second be added to the first, the sum will be 28.
Ans. The numbers are 20 and 24. Ex. 7. It is required to find two numbers such that their sum shall be 49, and the greater shall be equal to six times the less.
Ans. 7 and 42. Ex. 8. The sum of two numbers is 33, and if 5 times the less be taken from 3 times the greater, the remainder will be 35. What are the numbers ?
Ans. 8 and 25. Ex. 9. The mast of a ship consists of two parts ; one sixth of the lower part, added to one half of the upper part, is equal to 35 feet; and 3 times the lower part, diminished by 6 times the upper part, is equal to 30 feet. What is the height of the mast?
Ans. 130 feet. Ex. 10. Two persons, A and B, talking of their money, says A to B, give me one third of your money, and I shall have 110 dollars. Says B to A, give me one fourth of your money, and I shall have 110 dollars. How much had each?
Ans. A had 80 dollars and B 90 dollars.
EQUATIONS OF THE FIRST DEGREE CONTAINING THREE OR
MORE UNKNOWN QUANTITIES.
(115.) In the preceding examples of two unknown quantities, the conditions of each problem have furnished two equations independent of each other. In like manner, if a problem involve three or more unknown quantities, it must furnish as many independent equations as there are unknown quantities. Ex. 1. Take the system of equations
+ y+z= 9, (1.) 2x+3y+z=17, (2.)
6x+5y+z=31. (3.) In order to eliminate z, let us subtract equation (1) from equation (2); we thus obtain
x+2y=8. (4.) Also, subtracting equation (2) from equation (3), we obtain
4x+2y=14. (5.) We have now obtained two equations (4) and (5) containing but two unknown quantities, and we may proceed as in Section VIII. Subtracting equation (4) from equation (5), we find
Quest.-When a problem involves three or more unknown quanti. ties, how many equations must it furnish ?
x=2. Substituting this value of x in equation (4), we obtain
Substituting these values of x and y in equation (1), we obtain
z=4. These values of x, y, and z may be verified by substitution in the original equations.
(116.) We have effected the elimination in this case by method third, Art. 113; but either of the other methods might have been employed. Hence, to solve three equations containing three unknown quantities, we have the following