A Treatise on the Elements of Algebra (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1821 Excerpt: ...( fl ) +c'2/'=rf, 65 23 88 or 42 11 7 u= = =--or-24 24 3 " 4 Either value of v will answer the conditions of the question; 12 IS =z;-l_i and y=4, x=vy=-x4 = 7Hence, the numbers are 4 and 7. Ex. 3. Find two numbers, such, that the square of the greater minus the square of the.less may be 56; and the square of the lesser plus-yd their product may be 40. Answer, 9 and 5. Ex. 4. There are two numbers, such, that 3 times the square of the greater phis twice the square of the less is 110; and half their product plus the square of the lesser is 4. What are the numbers(a)? Answ. 6 and 1. XXVI. On the Solution of certain Equations, in which the Two unknown Quantities (x and y) are similarly involved. 89. Let x and y be any two numbers, of which x is the greater, andy the lesser; let x+y = 2s, x--y=2z; then, by Art.28, x=s + z, a.ndy=s--z. Now let x+y--a, x3+y3 = b, x+y=c, ?indxs+y5=d; then the values of x and y may be found (a) For a great variety of questions relating to quadratic equations which contain two unknown quantities, see Bland's Algebraical Problems, 181?. 121 CHAP. VI. ON RATIOS, PROPORTION, AND VARIABLE QUANTITIES. XXVIL Definitions. 92. By Ratio is meant the relation which one quantity bears to another, with respect to magnitude. It is evident that this relation can exist only between quantities of a similar kind; thus, a number must be compared with a number; a line with a line; &c &c.; and it would be absurd to compare a certain number of feet with a certain number of pounds &c. &c. 93. There are two ways in which the magnitude of quantities may be compared. In the first place, they may be compared with regard to their difference; and then the question asked, is, "How much one quantity is greater or less than another.&q...