than 36. Ifa=80,b=60, c=40, and d=20, how much more is ab+be+cd + ad (a−b) (c-d) 37. A boy expressed his age by saying, that if a=6, b=5, and c=7, he was a + ac 38. His grandfather said, "Well, at that rate, I must be How much older was he than his grandson? 33 abc 29 a2+b2+c2* In the addition of like quantities, to the difference of the sums of the positive and negative coefficients prefix the sign of the greater, and to it attach the proper letter or letters. Thus, 3a-126-7c2+5d+8e-f -5a-4b-5c2+7d-4e+m 8a+ 136-3c2-d+7e-z2 6a-3b-15c2+11(d+e)−ƒ+m−z2. The addition of unlike quantities is performed by merely arranging the different expressions in alphabetical order, and connecting them by their proper signs. The student is advised to make use of brackets as often as possible in expressing the sums of quantities. EXERCISE VI. 1. 2a+a. 2. 9a+15a. 3. 762+362. 4. 5a2b+9a2b-8a2b-3a2b. 5. 10d-7d. 6. x+13x-8x+7x-2x. 7. 5a-b+2c-4d+6e. 8. 5m-3m-2n. 9. 6(a+x)+5(a+x) −7 (a+x)+(a+x) −2(a+x). 10. (a−y)+12(a−y)—5(a−y)+2(a−y) — 9(a−y). 11. ax + bx. 12. 5a+4b-3c-7d+8+3a-126+7c-10d-4. 13. 12h-3c-7f+3g-3h+8c-2f-9g+5x. 14. 16a-5b+10c-9d+3a+18b-5c-7d+3e-7a-2b-3d+5e 9h+11a-3b+2c+8d+7h. 15. 12a2-6bc+8c2-f+13-7a2-3 bc+502+9f-9+5a2+18bc 11c2-3f+10+4bc-6c2+f-11. 16.4g8h-k+31+7+7g-5h+11k-l-3+g+h-12k+81 -10. 17. 15x-12y+3x-v-3x+4y-z+8v+x+72-v-3y-4x+9v2x+5v. 58. 2d. 18. 7p+3q-v+s+3p+5q−3v+7s−6p+q−s−6q+5v+3s+v+ 19. 6a-4b+3c+6b+4d-4c+3a+a-d+2c+2b+c-b+e+5a+ 20. 3a2b-7 ab2+xy+7xy-2a2b-ab2+2a2b+7y2+2b3+2a3—xy+ 3ab2-a3-763-xy-x2-7xy+2y2. 21. 3ax+4by+2x+7cx-3dy-5z+8cxy-z. 22. 3x+4y+3√√z+5√√/y-2/x-8√√x+10√2-8√√x+y+ √x-7√y+15%. 23. 12 ax3-17 a2x2+9a2x2-14a3x+7a3x—7—ax3—a1+3a2+12α3x+ 8a2x+9a3x-4x3-2a4. 24. xyz+x3-xx3+x2x+2x2y+5y3-7xy2+3x2-9 y3-x2z+8xy2 +7x2y+3y3+5x2+3xyz-7x2-8y3-3x2-xx2+z8—9x2z-2xyz+xz2 Rule. Change signs in the subtrahend, and then proceed as in addition. Suppose that b has to be taken from a following the rule, the expression will be written a—b, and the reason is clear. But supposing that b-c has to be taken from a, the subtrahend must become, according to the rule, b+c, the whole expression being a-b+c. And the reason of this, if not apparent at once, is evident after a little consideration. For as a is not to be diminished by b, but by b-c (that is by c less than b), so a-b and -c also, would be a diminution too great by c. Let a=8, b=6, and c=4, then a−b—8—6, −2 ; a−b—c =8-6-4, 8-10, -2; but a-(b-c) or a-b+c (with the sign changed) 8-6+4, or 12-6=6. = The reason for changing signs may also be illustrated thus: x= x-y+y; and if we subtract +y from x, x-y remains, just as if —y had been added to x according to the rule. Again, x=x+y-y, and if y be subtracted from x, x+y remains, as though +y had been added to it, according to the rule. Very often there will seem to be no necessity for changing signs in subtraction, since by simply deducting the smaller of two coefficients from the larger, and annexing the proper letter, the end will be answered; thus in taking 3a from 5a, the remainder is seen at a glance to be 2a; but in reality the rule still comes into operation, for 3a and 5a are both + in the first instance, and the sign of +3a must be changed into -3a before the result +2a is obtained. 1. Take - EXERCISE VII. 8a from -16a; -12ac from -18ac; -3ab from -11ab; -5xyz from -16xyz. 2. Take 5(a+d) from -16(a+d); -14(x+y+2) from -21(x+y +*); -15a(bc+de) from -32a(bc+dc); −a(b+c) from −3a(b+c). 3. How much is a more than -a? 2a more than -2a? 36 more than -36 4x more than -4x ? 4. How much is 5ac more than -5ac? 6abc more than -6abc? 7(x-y) more than -7(x-y)? 8(x+y+2) more than -8(x+y+2)? 5. How much is 16a more than -2a 32ab more than -15ab? ab more than -11ab? 3abcd more than -12abcd? 5(a+e) more than -12(a+e)? 16(x+2) more than -(x+)? 6. Take 3a+46 from 11a+56; 16x+y from 25x+16y. 7. Take 24xy+16z from 25xy+20%; 4ab+2ac from 16ab+3ac. 8. Take 4a from 0; 66 from 0; 16ac from 0; 32x from 0; 3abc from 0; 4(x+y) from 0; 5(a+b) from 0. 9. How much is 4a-(3a+b)? 16-(11b+5c)? 3x-(x+4y)? 2xy −(xy+yz)? 4xyz— (3xyz+5uv)? 10. Take -4 from 0; -a from 0; -b from 0; -6ab from 0; 7xyz from 0; 3(x+y) from 0. -3a from 0; 11. How much is 3a-(a-3b)? 6ab-(4ab-3c)? 16xy-(11xy-5≈)? 3xyz-(xyz-4a)? 5abc-(2abc-4d)? 12a2-(6a2-462)? 12. How much is the difference between a+b and a-b? 13. How much is the difference between 3a + 4b and 3a-4b? 14. Take a-b+c from 2a+b-c; 4b-3c from 36+4c; x+y-z from-x+y+z. 15. Take 3a from 56; 5x from 6y; 32 from 7x; 5ab from 3cd; 3ac from 5xy; -5xy from 3ab; -5x from 12xyz. 16. How much is 3a-(-4a)? 5b-(-3b)? 16ac-(-5ac)? 11xyz -(-3xyz)? 17. How much is a(x+y) more than b(x+y)? 4(a+b) more than x(a+b) 3a(b+c) more than 46 (b+c)? 5p(q+r) more than -39(q+r)? 6a(c+d) more than -5d(c+d)? 18. Subtract 5a-7b+3f from 11a-5b+5ƒ; and 11a-5b+5ƒ from 5a-7b+3f. 19. Subtract a+26-3c from 3a-2b+c; and 3a-2b+c from a+ 26-3c. 20. Subtract 5a+3c+5e-11g from 16a+5b-7d+3f; and -16a +5b-7d-3f from 5a+3c+5e-11g. 21. How much is 3x+5y-6z more than -3x-4y+6z; and -3x -4y+6% more than 3x+5y-6z? 22. From the sum of 3a-4b, 6a+2b, −3a+7b, and 5a-3b subtract the sum of 4a+6b, 3a-2b, and -12a+5b. 23. Required the sum of the difference of (7c+8d)-(12c-7d) + the difference of (12c+5d)-(8c-4d). 24. How much is (6c-4ef)—(12c+ef) greater than(8c+8ef)-(-3c +ef)? 25. Subtract the sum of (a+b)+(a—b) from (a+b)—(—a+b). 26. How much is (7x+12y)—(−8x+4y) greater than (3x+y)+ (4x-y)+(-12x+5y)? 27. How much is a-b less than a—(—b)? 28. If from a(x+y), b(x+y) is subtracted, and then again c(x+y) added, what is the sum ? 29. From 3a3-7a2b+4ab2—363 take 9a3-a2b+2ab2—2b3. 30. From 7xy+9y√y−11√xy subtract — 2x √xy+2x√y—y√y +12a. 31. From 13a√(x−y)−17b √(x+y)—axy subtract 176√(x−y)+ bxy-x2+a2y. 32. From the sum of 8a3-8a2+17, 8a2-18+7a, and 6a-a2-7a3 -11 subtract the sum of 10a2-3a-16, a—a2+a3, and 8a2+2a+18. It has been said that when several letters or combinations of letters go to form but one quantity, they are united by a vinculum, or placed within brackets. Thus x-y- -z is the same as x- -(y-x). Further instructions in the use of brackets will now be given. ab cd In the addition, subtraction, multiplication, and division of simple quantities, brackets are not needed, so long as the quantities are taken up in alphabetical order: e. g. instead of {(a+b)+c}+d, we write a+b+c+d; and instead of {(axb)+c}+d, we write ; but if the multiplicand or the dividend is a sum or a difference, the bracket must not be omitted. Thus it will not do to write a+b×c instead of (a+b)c, nor a−b÷c instead of a с Whenever a positive or negative sign precedes a quantity within brackets, the sign affects the whole quantity. If the whole quantity is to be added, precedes it, and the signs within the bracket remain unchanged; but if the whole quantity is to be subtracted, to be subtracted, precedes it, and the signs within the bracket must be changed. Thus +(x+y—z)= x+y-z, and (x2-2xy—y2)= x2 — 2 xy-y2; but −(x+y−z)=−x—y +z, and -(x2-2xy-y2)=-x2+2xy + y2. In the case of a double bracket, we have 3x-{(x-3)-(2y-z)}=3x-(x−3z)+(2y—z)=3x-x+3x+ 2y-z=2x+2y+2z. The preceding remarks are equally applicable to fractions with numerators of more than one term, as often as a change of form is necessary. Thus —****, [or —}(x+y−2)]=−−2+3 [or —дx−fy +1≈]; and x+y-z -(x-y), when multiplied by 4, becomes (x-y), or −x+y. EXERCISE VIII. Simplify as far as possible 1. (3b2+by)+(3a2-by)-(b2+a2). 4. {3d x-(5c-xy3z)}+(4c-8d2x-xy3z). 8. (364+8cyz)+{(7—64)—(7—864)} — (8cyz+6b4). 9. (1262e4+cǝ3) — {4m2p—(4b2^ — 6m2p)}+(8b2a—2cx3). —yz2). 10. (11a2bc4+3a3b2) — {20b3+(a3b2 — 4a2c)} - {2a3b2+(4a2c+3a2bc1)}. 11. {(3x2y2x2—2yz2)—(xyz—yz2)}-(2x2y2x2—xyz)—(x2y2x2 12. {56325-(4p2q2+m3n)} — {(20b31⁄23 — 1) − (6 m3n +16 b31⁄23) } — {p2q2 +(5m3n—3p2q2)}. It must be borne in mind that there are literal as well as numerical coefficients. mx2+nx2 —px2 + qx2 is an expression in which the several quantities m, n, p, and q, are coefficients of the common factor x2, or vice versâ. This is very conveniently shown by means of a bracket, thus, (m+n-p+q)x2; ax" + bx"+cx"+dx"=(a+b+c+d)x". EXERCISE IX. Collect coefficients in the following: 1. ax+bx+3a2x2+4b2x2. 2. ax3+bx3+cx3. 3. px2+qx2+ax+rx2+bx. 4. mxy+by2+nxy—pxy+dy2. 5. ax+ay+by-cx+bx-cy. 6. xyz+az3-xz3+x2x+x2y2+y3z-xy2. 8. px2y2-q2x2+ax2y2—a3xy2+pxy2. |