Solution. If we proceed to eliminate y between these two equations, by the process of art. 155, the remainder of the first division is (x2—6x+5) y2—(10 x2—60 x 50)y+24 x22—144x+120, in which x2-6x+5 is a factor of each of the coefficients of y, and y2, and of the terms which do not contain y.. Before suppressing this factor, we must see whether, as in art. 157, it may not be equal to zero, in which case we have is substituted in the given equations, each of them becomes y35 y2+6y=0, which is satisfied by the value is substituted in the given equations, each of them becomes which is the same as the preceding equation, and gives therefore the same values of y. Having thus obtained all the roots of the given equation corresponding to x2 · 6 x + 5 = 0, we may omit this factor of the above remainder, and it becomes y2-10y+24; and as this does not contain x it is unnecessary to proceed farther in the elimination of y, but we may obtain the roots of the equation which are y2-10y+24=0, and substitute them in the given equation to obtain the corresponding values of x. Thus, if the value y = 6 is substituted in the given equations, each of them becomes is substituted in the given equations, each of them becomes z=5, or =1, in either of which cases, y=0, or=2, or =3; or x = † (24±√171), in which case, y = = 6; or x= 2, in which case, y = = 4. - ✓ 15; + √ 15. or x= 7, y= +√15, z = or x 7, 21. What two numbers are they, whose sum is 32, and product 240? Ans. 12 and 20. 22. What two numbers are they, whose sum is a, and product b? Ans. a+v(a2—b), and a-√(‡a2—b). In what case would the values of these unknown quantities be imaginary? Examples of Equations of the Second Degree. Ans. When we have that is, ⋅ b > 1 a2, b > ({ a)9; that is, the product of two numbers cannot be greater than the square of half their sum. 23. What two numbers are they, whose difference is 5, and product 24? Ans. 8 and 3; or -3 and .8. --- 24. What two numbers are they, whose difference is a, and product b? Ans. a±√(b+† a2), and — 1a±√(b+}a2). 25. Find a number, whose square exceeds it by 306. Ans. 18, or 17. "My 26. A person being asked his age, answered, “ mother was 20 years old when I was born, and her age multiplied by mine, exceeds our united ages by 2500." What was his age? Ans. 42. 27. A person buys some pieces of cloth, at equal prices, for $60. Had he got three more pieces for the same sum, each piece would have cost him $ 1 less. How many pieces did he buy? Ans. 12. 28. A person dies, leaving children, and a fortune of $46800, which, by the will, is to be divided equally among them. It happens, however, that immediately after the death of the father, two of his children also die. If, in consequence of this, each remaining child receives $ 1950 more than it was entitled to by the will, how many children were there? Ans. 8. 29. Twenty persons, men and women, spent $48 at an inn; the men $24, and the women the same sum. Now, on inspecting the bill, it is found that the men have to pay |
