the greater. The answer must have the same sign as the greater quantity. Hence it appears, that what is called Addition in Algebra, is sometimes performed by Subtraction. This is an apparent contradiction, which is very apt to perplex beginners. The whole difficulty will vanish, however, if they remember the object of Addition; which is, to express the value of two or more quantities, of any kind, in the most simple manner possible. 23. What is the sum of 5 x y, and 3 x y? Let us suppose the positive quantities in this question, namely, 5 x and 4 x, to represent sums of money received, and the negative quantities, 3 x and -2x, payments made. Then, 9 x dollars are received, and 5 x dollars paid away; and, of course, 4 x dollars remain on hand. If the value of x be 100 dollars, then 900 dollars are received, 500 dollars paid away, and a balance of 400 dollars remains. 1 30. Add together 4 x, and 8 x, and 7 x, and 6 x. ANS.-5 x. This example is like the last; but here, the sum of the negative quantities is greater than that of the posi tive, and the difference is marked with the sign The sum of 4 x and 6 x is 10 x; 8 x and 7 x is 15 x; the difference, or balance between the two sums, is 5 x. If a trader gain 40 dollars in one week, lose 80 dollars the next, lose 70 dollars the third, and gain 60 dollars the fourth, in the four weeks he gains 100 and loses 150 dollars; that is, he loses 50 dollars more than he gains, and his amount of loss and gain stands 50. In this case, we suppose the value of x to be 10 dollars. 31. Add together 5 ab c,-7 abc, 3 abc,-abc, and 2 a b c. ANS. 2 a b c. The sum of the positiye quantities is 10 a b c; and that of the negative quantities is 8 a b c; and 10 a b c 8 a b c 2 ab c. = When several similar quantities, some of which are + and others —, are to be added, it will generally be found the most convenient way to proceed as in the last three examples, namely: Reduce them to two terms, by bringing all the positive quantities into one sum, and all the negative quantities into another, and then balance these two terms in the usual manner. 32. Add together 8 x, and - 3 x, and 4 x, and 40. Add together 14 a b x, and 6 a b x, and -7 a b x, and 3 a b x, and 7 a b x, and 5 a b x, Simple Quantities that are Dissimilar. 1. What is the amount of 4 a, and 3 x, and 5 y ? ANS. 4a+3 x + 5 y. In this question, the quantities are dissimilar, that is, different from each other; and, of course, they cannot be actually added. All that can be done is, to connect them together by their proper signs. 2. What is the sum of 3 b, and 7 c, and 5 x? ANS. 3b7c-5 x. 3. Add together 6 x, and - 5 y, and 7x y, and .4 z. 4. Add together 7 a b, and — -6. 4 y, and 12 bx, and 9. What is the sum of 4 a b, ard 5 a x, and - a b, and — 16? ANS. a xabc-16. 3 ab, As these In this question, we have +4 a b; also and — ab, which, together, make — 4 a b. 4 ab. two quantities, +4 a b and — 4 a b, cancel each other, they do not appear in the answer. We also have 5 ax and 6 a x, which balance each other in part, the difference between them being + a x, which is the first term of the answer. The remaining terms, + a b c and 16, being dissimilar, are annexed. a b 16, is a true In all cases, the ADDITION is performed, if we connect together the several given quantities by their respective signs. Thus, 4 a 5 a x ·3 ab + 6 a x + a b c answer to the last question, and expresses the value of the several quantities as perfectly as the answer given above. What remains, is a species of reduction, by which we unite together as many of the terms as we can, for the sake of diminishing their number, and rendering the answer more simple. It is evident that two equal similar quantities whose signs are unlike, as + 5 x and 5 x, cancel each other. 10. What is the sum of 16 a b, and 14. Add together 5 a b c, and 12 x, and and 4 a b c, and 16 x, and 3y, and and 76, and - 14 a x, 5 ax, a b c, and 5 x, and 83, and 16 a x. 15. What is the sum of 15 a x y, and a b, and 17. Add together 6 a b, 5 x y z, 14, — 8 a b, 21, 18. What is the amount of 8 x y, — 5 b c d, 18, -3xy, -9, b c d,- 12, and 4 b c d? |