completing the square, y2 - 8 xy + 16x = 16x + 64; 4x = ± 4√(x + 4), and y = 4x + 4 (x + 4). Also from the second equation, y - 2yx = 4 ; completing the square, y - 2yx + x = x + 4 ; extracting the root, y - x = ± √(x + 4); .. 4y = 4x + 4 (x + 4) = y, from the last equation; 5 5 45 √ + √y + √√√2 + √y += 10 +=; extracting the root, ✓ (√2 + √3) 5 4 4 √5 4 : √x + √y = 5, or 20, supposing the former, .. by involution, .. by subtraction, xy x (xy + xy) = 125xty - 570; or xy x xy x (x + y) = 125 xy - 570; or 5xy = 125 xy - 570; .. by subtraction, x - 2xy + y = 1, or - 51; extracting the root, - y = ± 1, or ± √(-51); but x + y = 5; ... by addition, 2x = 6, or 4, or 5 ± √ (- 51); by subtraction, 2y = 4, or 6, or 5 + √(- 51); ; - 13 + √(-51) * 2 * The other case, where √x+√y = 20, is solved in the same manner. SECTION VI. On the Solution of Problems which involve Simple Equations. (32.) THE solution of a problem, or method of discovering by analysis quantities which will answer its several conditions, is performed by assuming algebraic symbols to represent the quantities sought, and by deducing equations from the application of these, in the same manner as if they were known quantities, to the conditions of the problem. The independent equations derived from this process, if the conditions be properly limited, will equal in number the unknown quantities assumed; and from the solution of these several equations by the rules already given (23. 27. 29), the values of the algebraic symbols will be determined. Whether these values are correct, may be determined synthetically, by applying them instead of their respective symbols to the several conditions of the problem. If the conditions of the problem are not properly limited, that is, are not sufficient in number, or not sufficiently independent of each other, the resulting equations will either exceed in number the unknown quantities, and will therefore some of them be identical or inconsistent, or will be fewer in number than the unknown quantities, and consequently will admit of an indefinite number of answers. In many cases, instead of assuming a symbol to represent each of the required quantities, it is convenient to assume one only, and from the conditions of the problem to deduce expressions for the others in terms of that one and known quantities. And as the number of conditions ought to be one more than the number of quantities thus expressed, there will remain one to be stated in an equation; from which the value of the unknown quantity may be determined (22.27. 28): and this being substituted in the other expressions, their value also may be discovered. Examples of the Solution of Problems producing Simple Equations involving only one unknown quantity. 1. What number is that, to the double of which if 18 be added the sum will be 82? 2. What number is that, to the double of which if 44 be added, the sum is equal to four times the required number? Let the number. Then 2x + 44 = 4x, by supposition; .. by transposition, 44 = 2x and 22 Х. 3. What number is that, the double of which exceeds its half by 6? Let x = the number. |