Thus from 5x-3y+4z take -2x+4y+5z. When any term to be subtracted has no term in the minuend to correspond to it, it can only be written after the other terms with its sign changed. From 2a-36+5c take 3a+4b+d. (1) From 5x+3y+2z take 3x+2y+z. (2) From 7a+5b-3c take 11a+4b+2c. (3) From 5q-7r+4s take 4q+8r+5s. (4) From 8m-3n-4p take 10m+4n+2p. (5) From -4a+5b-10c take 5a+9b+3c. (6) From 2xy+5yz+8xz take 5xy-3yz- 2xz: (7) From 15x-12y+10z take -5x+10y-5z. (8) From 1-2ab+5bc7ac take ab+2bc-3ac-7. (9) From ax-by+cz take 1-3by — 2ax+cz. (10) From - 8p+5q-10r+4 take 4p+10q-20r -16. (11) From 2a-b-2c+2d take 3a+b-c-d, and from the result take -5a+2b+c−3d. (12) From xy-xz+10yz take 20xy+10xz — 10yz, and from the remainder take 2xy—3xz+5yz. When we have a number of like terms, some of which are to be added and some subtracted, we collect all those which have the plus sign into one sum, and likewise those which have the minus sign, according to rule (1), and then subtract the results by rule (3). Thus 2x+5x-9x-3x+x-7x+8x. 1st, taking all those to be added; and 2nd, those to Or all may be put into one column thus: + x If there are terms containing other common letters they are to be treated in the same way. The example (A) is still called addition, although it cannot be worked without a subtraction. The result -3a is called the algebraical sum of the terms 2x, 5x, &c. Find the algebraic sum of 5x-3y+2z, -3x+5y— 7z, 2x+4y-3z. The following arrangement will easily be understood: 5x-3y+2z -3x+5y-7z 2x+4y-3z 4x+6y-8z If a 1, y=2, z=3, the sum is EXERCISE V. Find the algebraic sum of (1)*7a-3b+5c, 2a+5b-3c, -5a-2b-9c, 12a +b-7c. (2) -9a+7b-c, 3a-2b+10c, -8a5b-7c, 4a-3b+2c. (3) 2ab-3ac+bc,-3ab+2ac-5bc, 4ab+6bc-3ac, 10ab-5ac+6bc. (4) 1-4a+b-3c,2a-10+3b+4c, 5b-a+5-3c, 12c+5a-106-8. (5) 8ab-1+4bc-5ac, 3ac+2ab-3-2bc, 5bc9ab2ac and 10-7ab+bc-ac. (6) a-b+c+d, b-a-c-d, c-a+b-d, d+a- (7) 10-5x+7y, 11+2x-3y, 20-15x+4y. (9) x-3,y+4, z-9,8+5x, 3+4y, 7+5%, 3x+4y, (10) xy+3xz+2, yz + xy-10, -2xz-3xy-20, xy+yz+xz. (11) 10x-y-z, 10y-x-z, 10z-x-y, -8x-Ey -8z. (12) xy+5, xz-8, yz+2, 5xz+5yz, 3xy-8xz, 2xy-yz, -3xz-3xy+10yz-100. *Find the numerical value of the first five results when a=1, b=2, c=3. Fnd the numerical value of the last five results when x=2, y=4 and z=0. Brackets. (a-b), {x+y}, [c+d-e], &c. Brackets are used to show that all the quantities enclosed within them are to be treated as one quantity. Thus (a+b) × c means that the sum of a and b is to be found, and the result multiplied by c. Also a+(b+c-d) means that (b+c-d) is to be added to a. The result is by the rule of addition a+b+c-d. N.B. A bar or vinculum a— b is sometimes used as a bracket. When an expression within brackets has the sign + before it, the brackets may be removed. Further, a-(b+c-d) means that (b+c-d) is to be subtracted from a, and the result by the rule of subtraction is a-b-c+d. If an expression within brackets has the sign before it, when the brackets are removed the sign of every term within the bracket must be changed. When there is more than one pair of brackets they may be removed in succession by the above rules; but the inside pair must be removed first. This direction must be observed at each step of the process. Ex. Simplify (a+x)+(a−x) - {2a-x-(x-2a)}. Taking away the inner bracket, enclosing (x-2a), the expression becomes (a+x)+(a−x) — {2α-x−x+2a} =(a+x)+(a−x) — {4a−2x}. We may now remove the first two brackets which have the sign + before them, but the third having the sign before it must have the signs of its terms changed. Therefore (a+x)+(a-x)-(4a -2x)=a+x +a-x-4a+2x=2x-2α=2(x—a). EXERCISE VI. Simplify the following expressions by removing the brackets and collecting like terms :— (1) (a-x)-(x−2a)+(2α-x) — (a−2x). (2) a-(b-c)-(a−c)+c-(a—b). (3) (a+b)-(5a+3b)-(2a-3b)-(b-a). (4) (a-b)-(a-c) − (b −c) + (a− d) − (b − a) + (d-c). (5) (x−2α)–(x − 2b) — { 2a − x − (2b+x)}. (8) {a — (b−c)} - {b− (e—a)} - {c— (a — L·) } — (a+b+c). (9) {1+x−(1−2x)} − (1 − x − (1 + 2x) } — {1— 1-2x}+(1+x). (10) Subtract 1—{1–[1− (1 − 4x)]} from 2-[x— 2-(2x-1)-(1-x-4x-1}] (11) (a− 2b+c) — { − (2a + b) — c — [2c − a + b 2b+2c-a]}-2c. (12) x- {y-[x-y-(x-y-y-x)]−x}. FACTORS AND POWERS. When two or more numbers are multiplied together the result is termed a product, and the numbers thus multiplied together are called factors. Thus 12=3×4. Hence 12 is the product of 3 and 4; also 3 and 4 are factors of 12. When all the factors of a product are the same, the result is termed a power of the factor. If the factor is taken twice, the result is called the 2nd power. If three times the 3rd power, and so on. Thus axa is the second power of a, and is written thus a2; a×a×a is the 3rd power of a, written a3, and so on; a5 is therefore the 5th power. The product of axb is written a.b or ab. It may be noticed here that a÷b is sometimes writ Find the numerical value of the following expressions when a=4, b=3, c=2, d=1 and x=12. (1) 5(5x-6)-4(4x-5)+3(3x-2)-(2x-16.) (2) 3(x-3)-2(x-2)+x−2(x+2)-3(x-1)+10. |