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and

and

and

and

xy-ax-ay+axy,

-2xy-2ay+3ax+15,
18 arx-73+13 a xy-am,
-15 a xy-13 am+43 + 18 arx,
arx-18ay-2axy+3am.

5. Add together the following quantities.

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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.

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The result then must be 16. But to perform the operation on the numbers as they stand, first subtract 13, which gives 22- 139. 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 13716, as required.

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First subtract b, which gives a-b

This quantity is too small by c because b is larger than b-c by 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.

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Operation.

a2x+3by-5 a c3 — 16 -3a2x-by+2 a c3 +22

-2 a2x+2by-3ac3+6

3bx2-7ax2+13
13bc-3 ax3-8.

Ans. 3bx-13bc-4ax+21.

17 ay +13 ay—a—3
2 ay-b-11a +5.

42 a xy-4 ax
17 ax-2axy-5

143-17 y
33+4y-16 ab.

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XIII. Multiplication of compound quantities is sometimes expressed without being performed. To express that a + b is to be multiplied by cd, it may be written a+bxc-d 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 (c — d) or (a+b) (cd). In the expression a+b (c-d), b only is to be multiplied by c

Multiply a+b by c.

d.

It is evident that the whole product must consist of the product of each of the parts by c.

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a+b.

20+4=24

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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 by 4.

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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.

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When both multiplicand and multiplier consist of several terms, each term of the multiplicand must be multiplied by each term of the multiplier.

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It is evident that if a + be taken c times and then d times, and the products added together, the result will be c+d times. a+b.

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16. Multiply ax-3ay+xy by 3ay+ax.

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In adding these two products, the quantity 3 axy occurs twice, with different signs; they therefore destroy each other and do not appear in the result.

17. Multiply

5ad3acd-5 ac

by

2 a2c+2ad.

18. Multiply 13 a3ry—2 aby2 + 3 c y3

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If 3b+2 c be multiplied by 2 a only, the product will be too large by 3 b times 36+2c; hence this quantity must be multiplied by 3 b, and the product subtracted from 6 a b +

4 a c.

This result may be proved by multiplying the multiplier by the multiplicand, for the product must be the same in both

cases.

23. Multiply 2 ad+3bc+2 by 4ab-2c.

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