Ex. 4. Given 2x+3y+5x= 61, x+4y+7z 66, to find x, y, and z. 3x+6y+8x=104, Ans. x=10, y=7, z=4. Ex. 5. Given 2x+6y+5x= 93, 4x+3y+8x= 95, to find x, y, and z. 5x+4y+9x=116, Ans. x=7, y=9, z=5. Ex. 6. Given x+y+x=29, (1) to find x, y, x+y+x=10, (3.) and z. Subtracting equation (1) from (2), we obtain y+22=33. (4.) Clearing equation (3) of fractions, we have 6x+4y+3x=120. (5.) Multiplying equation (1) by 6, we obtain 6x+6y+6z=174. (6.) Subtracting (5) from (6), we have 2y+3x=54. (7.) We have thus obtained two equations, (4) and (7), containing only two unknown quantities. Multiplying equation (4) by 2, we have 2y+4z=66. (8.) Subtracting (7) from (8), we have z=12. Substituting this value of z in equation (7), we obtain Substituting these values of y and z in equation (1), These values may be verified as in former exam. ples. Ex. 7. Given 2x+4y-3x=22, 4x-2y+5z=18, { to find x, y, and z. 6x+7y- z=63, Ans. x=3, y=7, z=4. Ex. 8. Given x+y+2x=34, 2x+ y− z=45, } to find x, y, and z. x-2y+ z= 7, Ans. x=20, y=8, z=3. Ex. 9. Given x+2y-3z=13, 3x+y+4x=51, to find x, y, and z. zx+y+1z= 7, Ans. x=9, y=8, z=4. Ex. 10. Given x+y+z=32, 1x+y+z=15, to find x, y, and z. x+y+÷z=12, Ans. x=12, y=20, z=30. Ex. 11. A market-woman sold to one boy 5 apples, 9 pears, and 10 peaches for 53 cents; and to another, 12 apples, 4 pears, and 6 peaches for 38 cents; and to a third, 8 apples, 11 pears, and 12 peaches for 66 cents. She sold them each time at the same rate. What was the price of each? Ans. An apple, 1 cent; a pear, Ex. 12. A market-woman sold at one time 12 eggs, 10 apples, and a pie for 35 cents; at another time, 10 eggs, 16 apples, and 2 pies for 40 cents; and at a third time, 18 eggs, 12 apples, and 4 pies for 66 cents. She sold these articles each time at the same rate. What was the price of each? Ex. 13. A grocer had three kinds of wine, marked A, B, and C. He mixed together 10 gallons of A, 4 gallons of B, and 2 gallons of C, and sold the mixture at $1.30 per gallon. He also mixed together 8 gallons of A, 4 of B, and 4 of C, and sold the mixture at $1.35 per gallon. At another time he mixed 3 gallons of A, 2 gallons of B, and 11 gallons of C, and sold the mixture at $1.50 per gallon. What was the value of each kind of wine? Ans. A was worth $1.20 per gallon; B was worth 1.40 C was worth 1.60 66 Ex. 14. The sum of the distances which three persons, A, B, and C, have traveled, is 300 miles. A's distance added to C's is equal to twice B's; and three times A's added to twice B's, is equal to seven times C's. What are their respective distances? Ans. A's is 120 miles; B's is 100 miles; Ex. 15. Three persons, A, B, and C, talking of their money, says A to B, give me half of your money, and I shall have $100; says B to C, give me one third of your money, and I shall have $100; says C to A, give me one fourth of your money, and I shall have $100. How much money had each? Ans. A had $64, B had $72, and C had $84. (117.) If we had four equations containing four unknown quantities, we might, by the methods already explained, eliminate one of the unknown quantities. We should thus obtain three equations between three unknown quantities, which might be solved according to Art. 116. We might proceed in a similar manner if we had five equations, or even a greater number. Either of the unknown quantities may be selected as the one to be first exterminated. It is, however, generally best to begin with that which has the smallest coefficients; and if each of the unknown quantities is not contained in all the proposed equations, it is generally best to begin with that which is found in the least number of equations. Ex. 16. Given x+y+z+ u=14, 2x+y+ z- u= 6, to find x, y, x-y+3x+4u=31, Ans. x=2, y=3, z=4, u=5. Ex. 17. Given x+y+2x=22, 2x+3u=33, to find x, y, z, and 3y+4z=43, 4y+5u=65, и. Ans. x=3, y=5, z=7, u=9. Ex. 18. A market-woman sold at one time 6 eggs, 10 apples, and a pie for 19 cents; at another time, 10 eggs, 24 apples, and 2 pies for 37 cents; at a third time, 12 eggs, 18 apples, and 14 pears for 41 cents; and at a fourth time, 12 apples, 6 pies, and 18 pears for 54 cents. She sold each article constantly at the QUEST.-How do we proceed with four unknown quantities? Which of the unknown quantities should be first exterminated? same price as at first. What was the price of each article? Ex. 19. A grocer had four kinds of wine, marked A, B, C, and D. He mixed together 2 gallons of A, 14 gallons of B, and 4 gallons of C, and sold the mixture at $1.42 per gallon. He also mixed together 10 gallons of A, 5 gallons of C, and 5 gallons of D, and sold the mixture at $1.45 per gallon. At another time he mixed 10 gallons of B, 4 gallons of C, and 2 gal1ons of D, and sold the mixture at $1.50 per gallon.. At another time he mixed together 6 gallons of A and 12 of D, and sold the mixture at $1.60 per gallon. What was the value of each kind of wine? Ans. A was worth $1.20 per gallon: B was worth 1.40 66 Ex. 20. It is required to find four numbers such that the sum of the first three shall be 30; the sum of the first, second, and fourth shall be 33; the sum of the first, third, and fourth shall be 35; and the sum of the last three shall be 37. Ans. 8, 10, 12, and 15. |