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+, the sign may be omitted before this term, but the sign must always be expressed. Great care is requisite in the use of the signs, for an error in the sign makes an error in the result of twice the quantity before which it is written.
Add together. 3a + 2bc - 3 and
5a - 3b + 2c and
7 ab + 4b c 8C and
-a + 36 — 2bc. The sum is 3a +260° -36 + 5 a— 360* + 2c +7 ab
+4bc-8C - a +36-2bc. But this expression may be reduced.
3a + 5a-a= 80-a=7a, and 260-3b0 +46¢ - 260= 660-560 = b c", and
- 3c +2c* -86* + 3 =-11c* +56=-6c; hence the above quantity becomes
7a +60 +7ab-6c. To reduce an algebraic expression to the least number of terms, collect together all the similar terms affected with the sign + and also those affected with the sign, and add the coefficients of each separately; take the difference of the two sums and put it into the general result, giving it the sign of the larger quantity.
Examples in Addition.
1. Add together the following quantities.
5 ab- 2 am and
3 ab-5 am + 2 am. 2. Add together the following quantities.
13 a n - 6 m+ *, and
76 m - 3 - 8 y, and
4 ans + 5 ax - 4y.
3. Add together the following quantities.
7 mab— 16 — 43 my, and
19 a cb-13 amb +37 may + 48, and
14 my-- 19 may + nb-nx, and
4nx - 3bn +23 a my-nb. 4. Add together the following quantities.
my- ax - ay+axy, and
- 2xy-2 ay + 3 ax + 15, and
18 arx--73 + 13 a xy-am, , and
- 15 a x y — 13 a m +43 + 18 a r x, and
arx--18 + ay-2 a xy + 3 am. 5. Add together the following quantities.
13 a x 2bx - 7, and
156 x - 17b xy + 16, and
47 acd-X, and 37-bc-2a + 43b y x, and
Subtraction of Compound Quantities.
XII. The subtraction of simple quantities, as has already been observed, is performed by giving the sign - to the quantity to be subtracted, and writing it before or after the quantity, from which it is to be taken. If it is required to subtract c+d from a +b it is plain that the result will be a +b-C
d, for the compound quantity c + d is made up of the simple quantities c and d, which being subtracted separately would give the above result. From 22 subtract 13–7.
13 7 = 6. and
22—6=16. The result then must be 16. But to perform the operation on the numbers as they stand, first subtract 13, which gives 22 — 13=9. This is too small by 7 because the number 13 is larger by 7 than the number to be subtracted, therefore in order to obtain a correct result the 7 must be added ; thus 22 - 13 +7= 16, as required.
From a subtract b
This quantity is too small byc because 6 is larger than bby the quantity c. Hence to obtain a correct result c must be added, thus ab + c.
This reasoning will apply to all cases, for the terms affected with the sign
in the quantity to be subtracted diminish that quantity; hence if all the terms affected with + be subtracted, the result will be too small by the quantities affected with - these quantities must therefore be added. The reductions may be made in the result, in the same manner as in addition. Hence the general
Rule. Change all the signs in the number to be subtracted, the signs + to, and the signs — to +, and then proceed as in addition.
Examples in Subtraction.
1. From a' x +36y- 5 a 6 - 16
Subtract 3a* x + by- 2 ac — 22
a' x + 3 by- 5 a 6 — 16 - 3 ao x — by + 2 ac + 22
Subtract 4. From
Subtract 5. From
2 a' x + 2 by- 3 a c + 6
Ans. 3 b x. - 13 bc-4a x + 21.
42 a x y - 4a x
143 - 17 y
Subtract 7. From
a + 3abc--1
Multiplication of Compound Quantities.
XIII. Multiplication of compound quantities is sometimes expressed without being performed. To express that a + b is to be multiplied by c-d, it may be written a +bxcd with a vinculum over each quantity, and the sign of multiplication between them; or they may be each enclosed in a parenthesis and written together, with or without the sign of multiplication ; thus (a + b) x (
cd) or (a + b) (c-d). In the expression a +b(c-d), 6 only is to be multiplied by c d.
Multiply a + b by c.
It is evident that the whole product must consist of the product of each of the parts by c. a+b
20 +4 = 24
1. Multiply 3 ab +2cd by ef.
Ans. 3 abef +2 cdef. 2. Multiply
5ac+bct 3cd by 2 e.
Ans. 10 acé +2bce'tőcde. 3. Multiply 6 a' b + 6* c by 3 a b*. 4. Multiply b c d + 52 ao % + 13 33 c d
7 a lc. 5. Multiply 2 abd + 3 abm + abw. by
3 a b x*. 6. Multiply a x + 3 a b x2 by 13 a b gris When some of the terms of the multiplicand have the sign they must retain the same sign in the product.
7. 8. Multiply a--b by c, also 23 - 5 hy 4. ab
= 18 4.
by 2 d.
20 = 72. Since the quantity a—b is smaller than a by the quantity b, the product a c will be too large by the quantity b c. This quantity must therefore be subtracted from a c.
9. Multiply 3 a b C 10.
2 ad +6d-3c by 5 a b. 11.
3bcd-ef- 2 ac by 12.
2 albe-5a +6 by 4 ao bé. 13.
17 acd-1+5 a' x - ab x
a' cd. When both multiplicand and multiplier consist of several terms, each term of the multiplicand must be multiplied by each term of the multiplier. 14. Multiply 12 + 5
by 7 +4.
84 +35 + 48 + 20
84 + 35 + 48 + 20 = 187 15. Multiply a + b
a + b ctd
ac + bc tad + bd. It is evident that if a + b be taken c times and then d times, and the products added together, the result will be c+d times