(10.) 3xy-4x3y3+xу5; x3у+3x3y3+3xу5; -3x3у+x3у3 — 6xy3. (11.) 1-x; 2x−2; x+4; 9-x. (12.) 3a2-b2; 2ab-2a2; 2b2- ab. Find the sums of the following : (13.) a2b2c2-4a6+6a1c2-c6; 6a2c4-b1c2+a2b1; b6-a2c1+7a6-9a2b2c2 +4b4c2; -3a2b1 — 9a1c2 + a2b2c2 - 3a2c4+11b2c4. Working. The coefficients are partly fractional, therefore the rules for the addition and subtraction of fractions must be employed. Hence we have for 1st column - 3+2+: 10+12+25 47 = 30 30 2nd column - + the positive 22. The principle upon which Subtraction is performed in Algebra is this:-Subtracting a positive quantity is the same as adding a negative quantity; and Subtracting a negative quantity is the same as adding a positive quantity. This may be understood from the illustration given in page 7: for taking a man's property to be a positive quantity, and his debts to be negative quantities, it is clear that to take 101. of property away is the same in effect as to add 107. to his debts; and to take away a debt of 107. is the same in effect as to add 10l. to his property. And generally to subtract + a is the same as to add Also to subtract is the same as to add - α - α + a Hence is derived the Rule for Subtraction of Algebraical quantities. RULE. Change the signs of all the quantities to be subtracted, and then proceed as in Addition. Ex. (1.) Subtract a+b-c from 2a+b+3c. Working. The quantities to be subtracted are +a, +b, and −c; changing their signs they become -a, -b, and +c; and writing them as now changed under the other line, we have to add together the two lines : 2a+b+3c a (10.) 3x3y-4x3y3+xy3; x3y+3x3y3+3xy3; −3x3y+x31⁄23 — Сxy3. (11.) 1-x; 2x−2; x+4; 9-x. (12.) 3a2-b2; 2ab-2a2; 2b2 — ab. Find the sums of the following : (13.) a2b2c2-4a6+6a1c2-c6; 6a2c4-b1c2+a2b1; b6-a2c1+7a6-9a2b2c2 +4b4c2; -3a2b-9a4c2+ a2b2c2-3a2c4+11b2c4. Working. The coefficients are partly fractional, therefore the rules for the addition and subtraction of fractions must be employed. Hence we have for 1st column - }+3+}= (21.) 10+12+25 47 = 30 And the whole sum is a2 962+ c2 -}a2 + b2 c2 22. The principle upon which Subtraction is performed in Algebra is this:-Subtracting a positive quantity is the same as adding a negative quantity; and Subtracting a negative quantity is the same as adding a positive quantity. This may be understood from the illustration given in page 7: for taking a man's property to be a positive quantity, and his debts to be negative quantities, it is clear that to take 101. of property away is the same in effect as to add 107. to his debts; and to take away a debt of 101. is the same in effect as to add 10l. to his property. And generally to subtract + a is the same as to add Also to subtract is the same as to add α + a Hence is derived the Rule for Subtraction of Algebraical quantities. RULE. Change the signs of all the quantities to be subtracted, and then proceed as in Addition. Ex. (1.) Subtract a+b-c from 2a+b+3c. Working.―The quantities to be subtracted are +a, +b, and −c; changing their signs they become -a, -b, and +c; and writing them as now changed under the other line, we have to add together the two lines : : Ex. (2.) From 6x take 3x+3y-3z-3. Changing the signs as before, we have 6x -3x-3y+3z+3 3x-3y+3z+3 = Remainder. Note.-At first the beginner should be very careful to write down the lower line with changed signs as in the above examples. After he is perfectly sure of the rule, the trouble of writing the lower line with changed signs may be spared: it will be sufficient for him to conceive the signs to be changed, and then to proceed as in addition. Examples III. (3.) From 2x+3y take x-y (5.) From x+y take-x-y (7.) From 2x+y-1 take y+1 (9.) From 23+1 take x2+1 (11.) From 3x take x2+3x-3 (13.) Subtract a-b-c from a+b+c (4.) From -6x+3y take -x+3y (6.) From -3x-4y take 3x-4y (8.) From 2+2 take 3x-2 (10.) From 4 take x+8 (12.) From 4 take -2. (14.) Subtract 3a3-5a2x-3ax2-3x3 from 5a3-4a2x+2x2 - 3x3 (15.) Subtract -2a3+4a4-6a5 from a3-4a4+6a5 (16.) Subtract a3-4a2x-ax2-x3 from 3a3-3a2x+ax2-2x3 (17.) Subtract 3y+3x23y3+3xy3 from 3x3y-4x3y3+xy3 Find the excess of the upper line over the lower line in the following (23.) (24.) (25.) (22.) Find the difference between a2+2a+3 and a +3 (26.) Find the difference between the sum of a+b and a−b; and the sum of 2a+3b and 5a-6b (27.) Subtract the difference of x and y from 4xy+x+y (28.) Subtract 5 times the square of y from 4y2+1 (29.) Subtract 4x2 - y3-4 from x2—y3 (30.) Subtract 1+4x3y+6x2y2+4xy3+y* from x1-4x3y +2y1 |