ALGEBRA. 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; x 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×a× b. 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 = 10, d=0. 1. Find the value of 5a+6b+2c+d+4a+7b+3c+5a+6b+2d =(5x4)+(6x8)+(2 × 10)+0+(4×4)+(7×8)+ (3x10)+(5x4)+(6x8)+(2x0) =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 (21 × 4)+(27x8)+(51 x 10)+(310) 84+216+510+0=810 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 x 12)-2(8×4)+(7x0)+6(4x0) +(5×10) =80+(5 × 96)-(2×32)+0+(6x0)+50 =610-64 = 546 6. Find the value of 4ac+15bd-7bce+6cd-4ab-7ae =4(8×12)+15 (10 × 4)-7(10 × 12×0)+6(12×4) -4(8×10)-7(8×0) = 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(10×4)-7 (4×0)+ 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 x 12 × 4)-17(8x0)+21(10 × 4) = (14x32)-(27 × 480)—(17x0)+(21 × 40) =1288-1296011672 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-3b2c+5c3d-d2c = 15(4×4) 5 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 [(4×4) (6×6)] + = (14x160)-(7x576)+(7x6400)-0 =2240-4032+44800 = 43008 12. Find the value of 3abcde+4a-e-3a2ed-7de =3(4x6x8x10x0) + 4 [(4×4) 0] — 13. Find the value of a2-b2-c2-d2-e2+ab-bc+de = (4×4)-(6×6)–(8×8)-(10 × 10)—(0x0)+ (4x6)-(6x8)+(10x0) ) 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, (3a-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. |