13. Add together 3 (x + y), 4 (x + y), and 7 (x + y). Ans. 6(x+y). 14. Add together 5 (2x+y-3 z), and 2(2x+y – 3 z). NOTE. If several terms have a common letter or letters, the sum of their coefficients may be placed in parenthesis, and the common letter or letters annexed; thus, 15. Add together ax-bx+3x, and 2ax +4 bx-x. Ans. (3 a +36 +2) x. 16. Add together by 3cy +5ay, and cy +4by - 2 ay. 17. Add together 2xy-axy, and 6xy-3axy. Ans. (84a) xy. 18. Add together 7 (3x+5y)+3a-6x+8 ab, 3 x +5(3x+5y) +7a-5ab, and 8x +2 (3 x + 5 y) Ans. 47 x 70y+3a. Τα 3 ab. 41. From what has gone before, it will be seen that addition in Algebra differs from addition in Arithmetic. In Arithmetic the quantities to be added are always considered positive; while in Algebra both positive and neg ative quantities are introduced. In Arithmetic addition always implies augmentation; while in Algebra the sum may be numerically less than any of the quantities added; thus, the sum of 10x and 8x is 2x, which is the numerical difference of the two quantities. SECTION V. SUBTRACTION. 42. SUBTRACTION in Algebra is finding the difference between two quantities. 1. John has 6 apples and James has 2 apples; how many more has John than James? Let a represent one apple, and we have 2. During a certain day A made 9 dollars and B lost 6 dollars; what was and B for the day? must be considered lar, it is required to the difference in the profits of A If gain is considered +, then loss OPERATION. take and letting d represent one dol 6 d from 9 d. Hence it appears that, as addition does not always imply augmentation, so subtraction does not always imply diminution. Subtracting a positive quantity is equivalent to adding an equal negative quantity; and subtracting a negative quantity is equivalent to adding an equal positive quantity. Suppose I am worth $1000; it matters not whether a thief steals $400 from me, or a rogue having the authority involves me in debt $400 for a worthless article; for in either case I shall be worth only $600. The thief subtracts a positive quantity; the rogue adds a negative quantity. Again, suppose I have $1000 in my possession, but owe $400; it is immaterial to me whether a friend pays the debt of $400 or gives me $400; for in either case I shall be worth $1000. In the former case the friend subtracts a negative quantity; in the latter, he adds a positive. Or, to make the proof general: 1st. Suppose + b to be taken from the result will be and adding b to ab we have which is, as before, equal to 2d. Suppose b to be taken from the result will be and adding + b to a b we have 3. Subtract b+c from a. OPERATION. a (b+c)=abc a+b b subtracted from a gives a — b; but in subtracting b we have subtracted too small a quantity by c, and therefore the remainder is too great by c, and (a - b) would be just so much too small, and the remainder sought is a- b+c. 43. By examining the examples just given it will be seen that in every case the sign of each term of the subtrahend is changed, and that the subsequent process is precisely the same as in addition; hence, for subtrac tion in Algebra we have the following RULE. Change the sign of each term of the subtrahend from + or to +, or suppose each to be changed, and then proceed as in addition. to In examples 1-7, the minuend remaining the same while the subtrahend becomes in each 3 less, the remainder in each is 3 greater than in the preceding. In examples 8-14, the minuend in each becoming 3 less while the subtrahend remains the same, the remainder in each is 3 less than in the preceding. In examples 15-21, both minuend and subtrahend decreasing by 3, the remainder remains the same. 21. From 17 a x + 20 a. Ans. . 43 y. Ans 194 b. 14 xy 18bc44xy take 25 bc Ans. 17 a x43 b c +58 xy-20 a. 22. From 384 x 74y+18 c take 118x+74y-27 c. Ans. 266 x 148 y + 45 c. 23. From 2 y3 x1 — 10 3 take x2+4 y3 — x1 — 4 x3. Ans. 2x-6x3-5 y3. 24. From 6 a by-4xy + 3x z take-4aby-3xz 25. From 2+2xy + y2 take 2-2xy + y2. 26. From 2+2xy + y2 take 2+2xy - y2. 44. The subtraction of a polynomial may be indicated by enclosing the polynomial in a parenthesis and prefixing the sign. |