5. x = √↓ (a2 + b2 — c2 — d2) — 3/ (8 bc) + br. Ans. x = 9. Ans. x7. Ans. x = 14. 6. y =√ (a2+b2) −√ (a2 −b2)+3/ (a3—b3). Ans. y=5'693562. 10. y = a b c d 4 a2-10bc+2 26+ c 2c+d y= = 121. √ a + b + c a + b = c a − b + c b + c = a 2 2 2 Ans. y 2 3.799671. 11. There is a certain expression consisting of four terms connected by the sign. The first term is the square of a, divided by b; the second is the square of a by b; the third is the square root of the excess of the square of a above the square of b; and the fourth is the reciprocal of the sum of the squares 12. An expression consists of three positive terms; the first is the square of x, the second is x with a coefficient equal to the sum of a and b; and the third is the product of a and b. What is the expression? of a, b, c. Write down the expression. ADDITION. 32. ADDITION is the method of connecting quantities together by means of the signs prefixed to them, and incorporating such as are like into one Unlike quantities cannot be incorporated, and their sum can only be expressed by writing them in succession, prefixing to each its proper sign. sum. Thus the sum of 3a and 4 a is 7 a; for the sum of 3 times any quantity a and 4 times the same will be 7 times that quantity, in the same manner as the sum of 3 pounds and 4 pounds is 7 pounds. Again, the sum of 3 a 2b and 7 a 3b is 10 a sum of 3 a and 7a is 10a; but the first quantity 3 a 3a by 2b, and the second quantity 7a3b is less than 7a by 3b; therefore the sum of 3 a and 7 a, viz., 10 a, must be diminished by the sum of 2 b and 3 b, or 5 b, to obtain the correct sum of the two proposed quantities. 33. When the like quantities have like signs, their sum is obtained by adding the coefficients together, prefixing the common sign to that sum, and annexing the common quantity. It is usual to arrange the quantities which are like in the same vertical column, which facilitates the process of summation. 34. When the like quantities have different signs, add all the positive coefficients together, and then all the negative coefficients: subtract the less sum from the greater, and prefix the sign of the greater, and annex the common letter or letters. In the left column of example (5), the sum of the positive coefficients is 11, and the sum of the negative coefficients is 8; subtracting 8 from 11, the remainder 3 is positive, and hence the sum of the quantities in the first column is 3 a, the sign + being understood. In the second column the sum of the positive coefficients is 12, and the sum of the negative 14; subtracting the former from the latter gives 2, to which the sign is to be prefixed, and the sum is 2 a x2. This is obvious, since the sums of the positive and negative terms are 12 a x2 and 14 ax respectively, and the quantity to be subtracted exceeds the quantity to be added by 2 a x2, and this must therefore be written, not with the sign of addition, but with the sign of subtraction, prefixed. EXAMPLES FOR PRACTICE. 1. Find the sum of a + b + c, a + b + c. Ans. 2a+2b+ 2c, or 2 (a+b+c). 2. Find the sum of 2 a x + 3by, 3 a x + 2by, 7 ax + by, and 8 ax+7by. Ans. 20 a x + 13by. - 3. Find the sum of 2 a2 17 ab + 3b, 5 a2 + 12 a b 5 b2, 12 a2 +6 ab - 9 b2, 3 ao + 6 a b + 3 b2. Ans. 22 a7ab8b. 3 4. Find the sum of x-y3 + 2 x y2 - 3x2 y, 2x3-3x2-5 x2 y +2y3, 6 x2 y + 6x y2 — x3 — y3, and 5 x y 2 y3 - 4x3 +8xy. Ans. 2x+6x y + 10 x y2 - 2y". 4x + 7+ 8, 11 z 5. Find the sum of 2x + 3y-4 z-10, 8 y +5x — 10 y − 2, and 16 + 10x + 12y + 14z. Ans. 13x + 13 y + 28 z + 12. 6. Find the sum of 3x + 2 y3 + z3 + 8 x y z, y3 + 2 x − 3 1⁄23 — 4 xyz, 2+3x-2y3-2xyz, and x y z - + 2 + y3 + 23. 7. Find the sum of x + 3 x' y + x2 z 2 x v, 30 x1 - 29 x2 z +18 xv 17 x3y, 22 x3-15x - 32x2 + 16 xv, and 17x2 z - 12x+6xy-11 x v. Ans. 4 x14x3y - 43 x2 z + 21 x v. 2 35. If the coefficients be literal instead of numerical, they may be summed, when they are like, by the preceding methods; and, when they are unlike, their sum can only be expressed by writing them in succession, prefixing to each its proper sign. In this case annex the common quantity to the sum of the literal coefficients inclosed in brackets. Thus the sum of a x, bx, cx, dx may be written either ax + bx + c x + d x, or (a + b + c + d) x. 8. Add together a x · y, and (a 1) x + (b + 1) y. 9. Add together (a + 2b) x + (4b-3a) y, (2 a + b) x + (2 a−b) y, and 4 a x + 3by. Ans. (7a+3b) x + (6ba) y. 10. Add together px+qy+rz-c, 2px-2qy + 2c, 3 qy -px + 4c, and 7px-8qy rz-3c. Ans. 9px 36. SUBTRACTION is the method of finding the difference of any two quantities of the same kind. When the quantities are unlike, their difference can only be indicated by writing the quantity to be subtracted after the other, interposing the sign, which indicates subtraction. Let it be required to subtract 2x + 3y from 7x+6y. Here, if from 7x+6y we subtract 2x, there remains 5x + 6y, and if from this we subtract 3y, the result is 5x+3y, which is the difference of the two proposed quantities. Again, let it be required to subtract c d fron a from a b, c be subtracted, the remainder will be a-b-c; but, by subtracting c from a — b, we have subtracted too much, since c d is a quantity less than c by d. Having therefore subtracted too much from a b by d, it is evident that the remainder, a - b c, must be too little by d, and must hence be increased by d. The correct remainder is therefore a − b − c + d, or a − c − b + d, and if we inspect the signs of c―d, α -b-c+d or a cb ·† d 136 from 16 a - 9b. 9b the quantity to be subtracted, it will be seen that the signs are changed, the one from + to, and the other from to +. Lastly, let it be required to subtract 12 a By the receding example the reminder would be 16 a - 12a9b +13b, which, by incorporation, gives 4 a + 4b. This is the same result that would be found by changing 12 a into 12 a; - 13b into + 136, and then adding the columns. 16 a 12 a 13b 4a+ 4b, or 4 (a + b). Hence to subtract one quantity from another, conceive the sigu, or signs of the quantity to be subtracted to be changed, and proceed as in addition. In many cases it will be unnecessary to apply this rule of changing the sign, because the difference of the coefficients may be found at once, as in common arithmetic, and the common quantity annexed will complete the difference. Thus, in the preceding example, it is unnecessary to change the sign of 12 a, because 12 taken from 16 leaves 4, and therefore 4 a is the difference. So also, if - 7y be taken from 10y, the remainder is 3y. When the coefficients are literal, the operation of subtraction must be performed upon these coefficients, and to the remainder, enclosed in brackets, if necessary, annex the common quantity. 7. From 219 a3 -- √xy+ √z 3x2-2√xy + 2√√z 117 ab+218 a l2 + 145 63 subtract 193 a3 + 157 a b2 – 121 a2 b + 155 b3. Ans. 26 a3 + 4 a2 b + 61 ab2 – 10 b3. 8. Subtract 2 x3-3x2 y + xy2+xy3 from 5 x3 + x2y-6x y2 + y3. Ans. 3x+4x2 y 7 x y2+ y3 — x y3. y2 9. Subtract x4 4x3 y + 6 x2 y2 - 5 x y3 + y2 from 3x4 + 7 x2 y2 - 4 x y3 + y*. Ans. 2x4+3x3 y + x2 y2 + xy3. 10. From 2px2 - 3 qxy + ry subtract px2-4qxy+2ry. x2 + 4 a2 x + 8 a x2 + 4 a2x2 + 2 a x3. 37. To indicate the subtraction of a polynomial quantity, without actually performing the operation, enclose the polynomial in brackets, and prefix the sign 2 x3-3x2y + 2 x y2 — (x3 + y3 — x y3) signifies that the quantity 3+ y3 xy is to be subtracted from the preceding quantity. By the principle of subtraction, we must change all the signs of 3 + y3 x y2 from + to -, and vice versâ, and proceed as in addition. Hence the preceding expression becomes 2x3-3x2y + 2 x y2 — x3 — y3 + xy" = x3 — 3x2y + 3 x y2 - y3. Also, 12 a and a2+2ab+b2 - (a2 - 2 ab + b2) = a2+2ab + b2. a2+2ab- b2 = = 4 ab. On this principle we can make polynomials undergo several transformations, which are useful in various calculations. Thus, a2 2 a b + b2 = a2 — (2 a b − b2) — b2 — (2 a b — a2); a3-3a2b+3a b2 — b3 = a3 — (3ab 3 ab2 + b3)`' = a3 +3 a b2 (3ab + b3) = a3 b3 (3 ab 3 a b2). 12. Reduce 3 x — { (x − 3 a) — (2ya)} to its simplest form. Ans. 2x + 2y + 2a. 13. Reduce a2 - (b2 — c2 ) — { b2 — (c2 — a2) } + c2 − ( b2 — a2) to its simplest form. Ans. a2 3b2+3 c2. 14. To what is x + y + z − ( x − y) − ( y − z ) − (− y) equal? Ans. 2y+2z. 15. From a (x + y) − b xy + c (xy) subtract 4 (x + y) +(a+b)xy-7(x − y). Ans. (a4) ( x + y) − ( a + 2 b) x y + (c + 7 ) ( x − y). 38. We may now make a remark on the occurrence of a negative quantity in a detached form, and unconnected with a positive quantity. If 8 is to be subtracted from 6, it cannot be performed, in the arithmetical sense, since 8 is greater than 6 by 2: subtracting 6+2 from 6 by the principle of algebraic subtraction, we get 2 for remainder, implying that 2 still remains to be subtracted from some other positive quantity. In this way the student will notice that -7 a is greater than -11 a, because if a positive quantity, for example 14 a, be added to each the sum of 14 a and 7 a is 7 a, but the sum of 14 a and -11 a is 3 a; and the former sum is greater than the latter; therefore - 7a is greater than 11 a. In a similar manner it can be shown that O is greater than any negative quantity. For if we subtract successively 1, 3, 5, 8, 10 from 5, we get the remainders 4, 2, 0, 3 and 5, each of which, except the first, is evidently less than the one preceding it, so that 3 is less than nothing, and greater than MULTIPLICATION. 5. 39. MULTIPLICATION is the method of finding the product of two or more quantities, and the quantities themselves are termed factors. Let it be required to multiply a b by c. Here the product of a and c is ac; but the product of a less than the product of a and c by the product of b and c, that is, by be; hence the product of a — b and c is a c Again, if it be required to multiply a -b by c-d, d be called x; then we have, (ab) (cd) = (ab) x = ax-bx = a (c — d) — b (c — d) = a c a-b c-d ac-be -ad+bd ac-bc-ad+bd. ad- (bc-bd) |
