An Introduction to 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. 1818 Excerpt: ...shall have the least value of y=4, and the corresponding, or greatest value of x-Q2. And the rest of the answers will be found by adding 21 continually to the least value of y, and subtracting 17 from the greatest value of x; which being done, we shall obtain the six following results: x--Q2 I 75 I 58 I 41 I 24 I 7 y=4 I 25 I 46 I 67 J 88 I 109 These being all the solutions, in whole numbers, that the question admits of. Note 1. When there are three or more unknown quantities, and only one equation by which they can be determined, as ax + by + cx=d, it will be proper first to find the limit of the quantity that has the greatest coefficient, and then to ascertain the different values of the rest, by separate substitutions of the several values of the former, from 1 up to that extent, as in the following question. 5. Given 3x + 5y + 7=100, to find all the different values of ar, y, and z, in whole numbers. (0) Here each of the least integer values of x and y are 1 by the question; whence it follows, that r (0) If any indeterminate equation, of the kind above given, has one or more of its coefficients, at c, negative, the equation may be put under the form in which case it is evident that an indefinite number of values may be given to the second side of the equation, by means of the indefinite quantity z; and consequently, also, to x and y in the first. And if the coefficients a, b, c, in any such equation, have a common divisor, while /has not, the question, as in the first case, becomes impossible. 7 7 Consequently x cannot be greater than 13, which is also the limit of the number of answers; though they may be considerably less. By proceeding, therefore, as in the former rule, we shall have 100--5---7% l--2y--z x= 2----=33--v--2z----=wh. j And y=3p + z...