In Algebra quantities are represented by letters; the earlier letters of the alphabet, a, b, c, d, e, &c., generally standing for known quantities; and the later, x, y, z, &c., for unknown quantities. The same signs are employed as in Arithmetic, namely, + plus, for addition; - minus, for subtraction; × multiplied by, for multiplication; divided by, for division. When a number precedes a letter or symbol, it means this number multiplies the quantity after it; thus 5a = 5×a, 5ab=5× ab=5× axb. In taking the sum of quantities, partly minus and partly plus, that is, with unlike signs, it is best to sum up all the positive or quantities separately, then do the same with the negative quantities, and add the results thus separately obtained; thus, 4+3-2+3-7+8+7-8 When a = 4, b = 8, c = 1. Find the value of = 10, d=0. 5a+6b+2c+d+4a+7b+3c+5a+6b+2л =(5x4)+(6x8) + (2 × 10)+0+(4×4)+(7x8)+ (3x10)+(5x4)+(6x8)+(2×0) 20+48+20+0+16+56 +30+20+48+0=258 2. Find the value of 6a+226 +29c+19d = (6×4)+(22 × 8) + (29 × 10)+(19×0) 3. Find the value of 47a+246 +16c+12d = (47×4)+(24 x 8)+(16 x 10)+(12×0) 4. Find the value of 21a+27b+51c+31d = (21x4)+(27×8) + (51 x 10)+(31x0) When a = 8, b = 10, c = 12, d = 4, e = 0. 5. Find the value of ab+5ac-2ad+7e+6de+5b = (8×10)+5(8×12) - 2(8×4)+(7×0)+6(4×0) +(5×10) =80+(5 × 96)-(2×32)+0+(6x0)+50 6. Find the value of 4ac+15bd-7bce+6cd-4ab-7ae =4(8×12)+15 (10×4)-7(10x12x0)+6(12×4) -4(8x10)-7(8x0) =384+600-0+288-320—0 = 952 7. Find the value of 7abd+15b-8bd-7de +14ab-8bcd+4ac =7(8×10 × 4)+(15 x 10) -8(10x4)-7(4x0)+ 14 (8×10)-8 (10 x 12 x 4)+4(8×12) =2240+150—320—0+1120-3840+384 =3894-4160=-266 = 8. Find the value of 14ad-27bcd-17ae +21bd 14(8×4)-27(10×12 × 4)-17(8x0)+21(10 × 4) = (14x32)-(27 x 480)-(17x0)+(21 × 40) =1288-12960 = -11672 If a = 1, b = 2, c = 3, d = 5, e = 0. Find the values of (1) 2a+3b-2c+3d-4e+4a. Ans. 21. Ans. 24. Ans. 2. Ans. 11. Ans. 22. Ans. 4. Ans. -36 When a quantity is squared or multiplied by itself, this is expressed by a small figure called an index placed to the right of the quantity, and a little above it. Thus 22 means 2 × 2; a2 means a xa, and so on. When a number is cubed or multiplied by itself so that there are 3 terms, as 2 × 2 × 2, this is expressed by a small figure called an index placed as before, but the figure this time is 3. Thus 23 is 2 cubed, or the third power of 2; a3 is a xaxa, or the third power of a. Similarly a means a raised to the fourth power, or axaxaxa, and so on. When a = 4, b = 6, c = 8, d= 10, e = 0 9. Find the value of 15a2-12bc-352c+5c3d-d°c = 15(4×4) 12 (6×8) — 3 [(6×6)×8]+ 10. Find the value of 3a2-7ab2+6a2b2 — d2 =3(4x4)-7[4(6×6)]+6[(4×4) (6 × 6)]—(10 × 10) =(3x16)-7(4×36)+6(16 × 36)-100 =48-1008+3456-100 = 2396 11. Find the value of 14a2d-7a2b2+7c2d2-8e - = 14 [(4×4) 10] 7 [(4x4) (6×6)] + = - = (14× 160) — (7 × 576) + (7 × 6400)—0 12. Find the value of 13. 3abcde+4a3e-3a2ed - 7de Find the value of a2-b2-c2-d-e+ab-bc+de = (4×4)-(6×6)-(8×8)-(10×10)-(0x0)+ (4x6)-(6x8)+(10x0) 16-36-64-100-0+24-48+0 =40-248=-208 · When a quantity is enclosed within brackets ( ) or [ ], it means that all the quantities are to be considered as one: thus, (8+12) means 20, (12-8) means 4, (8-12) means-4, (a+2a+c) means 3a +c, (3α-6a+c) means -3a+c, − (6a+3a−b) means (9a—b), which is -9a+b. Note here that when a minus sign precedes a bracket, it changes all the signs of the quantities, as in the last example; so with the sign before a bracket we can remove the bracket without affecting the equation, but if we remove the bracket having a negative sign before it, we must change all the signs. |