Examples of Binomial Equations. 226. EXAMPLES. 1. Solve the two equations xy72y7-4 y3-8x+16=0, x2 y7 — 4 y7 — 4 x y3 + 8 y3 + 32 x — 64 — 0. Solution. The elimination of y between these two equations, by the process of art. 155, gives being substituted in the first of the given equations, produces 4 y74 y3 0, = which is satisfied by the value of y, y=0; or if we divide by 4 y3, we have 4 y4-10, y1 = 1, y=√1=±lor =±√−1, as will be shown when we treat of the theory of equations. Again, the value of x, X=- 2, being substituted in the first of the given equations, produces whence we have — 4 y3 +32= 0, y3 = 8, y= 2 or 1±√3, =— as will be shown in the theory of equations. Examples of Binomial Equations. 10. What number is it, whose half multiplied by its third part, gives 864 ? Ans. 72. 11. What number is it, whose 7th and 8th parts multiplied together, and the product divided by 3, gives the quotient 298? Ans. 224. 12. Find a number such, that if we first add to it 94, then subtract it from 94, and multiply the sum thus obtained by the difference, the product is 8512. Ans. 18. 13. Find a number such, that if we first add it to a, then subtract it from a, and multiply the sum by the difference, the product is b. Ans. (a2b). 14. Find a number such, that if we first add it to a, then subtract a from it, and multiply the sum by the difference, the product is b. Ans. √(a2+b). 15. What two numbers are they whose product is 750, and quotient 3? Ans. 50 and 15. 16. What two numbers are they whose product is a, and quotient b? α Ans. ab and ✔ Τ 17. What two numbers are they, the sum of whose squares is 13001, and the difference of whose squares is 1449 ? Ans. 85 and 76. 18. What two numbers are they, the sum of whose squares is a, and the difference of whose squares is b? Ans. (a+b) and √ † (a — b). 19. What two numbers are to one another as 3 to 4, the Ans. 342 and 456. sum of whose squares is 324900? 20. What two numbers are as m to n, the sum of whose CH. V. S v.] BINOMIAL EQUATIONS. Examples of Binomial Equations. 23-0159 21. What two numbers are as m to n, the difference of 22. A certain capital is let at 4 per cent.; if we multiply the number of dollars in the capital, by the number of dollars in the interest for 5 months, we obtain 1170413. What is the capital? Ans. $2650. 23. A person has three kinds of goods, which together cost $5525. The pound of each article costs as many dollars as there are pounds of that article; but he has one third more of the second kind than he has of the first, and 3 times as much of the third as he has of the second. How many pounds has he of each? Ans. 15 pounds of the first, 20 of the second, and 70 of the third. 24. Find three numbers such, that the product of the first and second is 6, that of the first and third is 10, and the sum of the squares of the second and third is 34. Ans. 2, 3, 5. 25. Find three numbers such, that the product of the first and second is a, that of the first and third is b, that of the second and third is c. and 26. What number is it, whose third part, multiplied by its square, gives 1944? Ans. 18. 27. What number is it, whose half, third, and fourth, multiplied together, and the product increased by 32, gives 4640? Ans. 48. 28. What number is that, whose fourth power divided by fth of it, and 167 subtracted from the quotient, gives the remainder 12000? Ans. 11. Cases of imaginary Solution. 29. Some merchants engage in business; each contributes a thousand times as many dollars as there are partners. They gain in this business $2560; and it is found that this gain is exactly half their own number per cent. How many merchants are there? Ans. 8. 30. Find three numbers such, that the square of the first multiplied by the second is 112; the square of the second multiplied by the third is 588; and the square of the third multiplied by the first is 576. Ans. 4, 7, 12. 227. Corollary. When the solution of a problem gives for either of its unknown quantities only imaginary values, the problem must be impossible. 228. EXAMPLE. In what case would the value of the unknown quantity in example 13 of art. 226 be imaginary? and why should the problem in this case be impossible? that is, when the product of the sum and difference is required to be greater than the square of a. required number is x, this product is (a + x) (ax) = a2 — x2; and, therefore, less than a2. Now if the R. |