+3y+5y—7y=(3+5)y—7y=(8—7)y=+y; -3+30-8-30-(8+3)=30-11=+19;

2x2 — 3x2 — 2x2 +6x2 = 8x2 — (3+2) x2 =(8—5) x2=+3x2;

5y3-3y2 -2y=5y2 — (3+2) y2=(5—5) y2=0 Xy2=0;+2x=2x.

When quantities with literal coefficients are to be added together; such as mx, my, px2, qy2, &c. (where m, n, p, q, &c., may be considered as the coefficients of x, y, x2, y2, &c.) it may be done by placing the coefficients of like quantities one after another (with their proper signs), under a vinculum, or in a parenthesis, and then, annexing the common quantity to the sum or difference.

Ex. 4.
ax+by+ b
bx+dy+2b

(a+b)x+(b+d)y+36

Ex. 5.

ax3+bx2+cx
ex3-dx2-fx

(a+e)x3+(b−d)x2+(c−ƒ)x

In Ex. 4. The sum of ax and bx, or ax+bæ, is expressed by (a+b)x; the sum of +by and +dy, or +by+dy, is =+(b+d)y.

37

In Ex. 5. The sum of ax3 and ex3, or ax2+ex, is =(a+e)x3; the sum of +bx2 and -dx2, or +bx2-dx2, is (b-d)x"; and the sum of +cx and -fx, or +cx-fx, is =+(c-f)x. Any multi

2

nomial may be expressed in like manner, thus; the multinomial mx2+nx-px2-qx2 may be expressed by (m+n-p-q)x3; and the mixed multinomial pxy+qy3—rxy+my―nxy, by (p-r−n) cy + (q+m)y2; &c.

Ex. 6. Add 2x2+y+9, 7xy-3ab-x2, 4xy-y -9, and xy-xy+3x2 together.

Ans. 4x+y2+10xy-3ab-y+x2y. Ex. 7. Add together 72a2, 24bc, 70xy,-18a2, and -12bc. Ans. 54a+12bc+70xy. Ex. 8. What is the sum of 43xy, 7x2,-12ay, -4ab,-3x3, ond -4ay?

Ans. 43xy+4x2-16αy--4ab. Ex. 9. What is the sum of 7xy,-16bc,-12xy, 18bd, and 5xy? Ans. 2bc. Ex. 10. Add together 5ax,-60bc, 7ax, -4xy, -6ax, and -12bc. Ans. 6ax-72bc-4xy. Ex. 11. Add 8a2 x2-3ax, 7ax-5xy, 9xy-5ax, and xy+2a3x together. Ans. 10a2x2-ax+5xy. Ex. 12. Add 2x2-3y2+6, 9xy-3ax-x2,4y3 — y-6, and xy-3xy+3x2 together.

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Ans. 4x2+y+6xy-3ax-y+x2y. Ex. 13. Add 3x2—4x3+x2, 5x2y—ab+x3, 4x2 -x2, and 2x3—3+2x together.

1

Ans. 4x3-x+5x2+5x2y—ab—x3—3. Ex. 14. Required the sum of 4x2+7(a+b)2, 4y2 —5(a+b)2, and a3—4x2—3y2—(a+b)2.

Ans. a3+x+y2+(a+b)2. Ex. 15. Required the sum of ax1-bx3+cx2, bxc2-acx3-x2x, and ax2+c—bx. Ans. ax4-(b+ac)x+(c+be+a)x2+(c2+b)x+c. Ex. 16. Required the sum of 5a+3b-4c, 2a5b+6c+2d, a-4b-2c+3e, and 7a+4b-3c-6e. Ans. 15a-2b-3c+2d-3e.

62. Subtraction in Algebra, is finding the difference between two algebraic quantities, and connecting those quantities together with their proper signs: the practical rule for performing the operation is deduced from the following proposition..

63. To subtract one quantity from another, is the same thing as to add it with a contrary sign. Or, that to subtract a positive quantity, is the same as to add a negative; and to subtract a negative, is the same as to add a positive.

Thus, if 3a is to be subtracted from 8a, the result will be Sa-3a, which is 5a; and if b-c is to be subtracted from a, the result will be a-(b-c), which is equal to a-b+c: For since, in this case, it is the difference between b and c that is to be taken from a, it is plain, from the quantity b-c, which is to be subtracted, being less than b by c that if b be only taken away, too much will have been deducted by the quantity c; and therefore c must be added to the result to make it correct.

it

This will appear more evident from the following consideration; Thus, if it were required to subtract 6 from 9, the difference is properly 9-6, which is 3; and if 6-2 were subtracted from 9, is plain, that the remainder would be greater by 2, than if 6 only were subtracted; that is, 9-(6-2) =9-6+2=3+2=5, or 9-6+2=9-4-5.

Also, if in the above demonstration, b-c were supposed negative, or b-c-d; then, because c is greater than b by d, reciprocally c-bd, so that to subtract from a, it is necessary to write atd.

64. The preceding proposition demonstrated after the manner of Garnier.

Thus, if b-c is to be subtracted from the quantity a; we will determine the remainder in quantity and sign, according to the condition which every remainder must fulfil; that is, if one quantity be subtracted from another, the remainder added to the quantity that is subtracted, the sum will be the other quantity. Therefore, the result will be a— b+c, because a-b+c+b-c=a.'

This method of reasoning applies with equal facility to compound quantities: in order to give an example; suppose that from 6a-3b+4c.

we are to subtract,

5a-5b+6c;

designating the remainder by R, we have the equality,

R+5a-5b+6c6a-3b+4c:

which will not be altered (Art. 49) by subtracting 5a, adding 56 and subtracting 6c, from each member of the equality; therefore, the result will be,

R=6a-3b+4c-5a+5b-6c,

or, by making the proper reductions,

R=a+2b-2c.

65. Another demonstration of the same proposition in Laplace's manner. Thus, we can write,

a=a+b-b

(1),

a—c=a-c+b-b.. (2);

so that if from a we are to subtract +b or —b, or which is the same, if in a we suppress +b, or -b, the remainder, from transformation (1), must be a-b in the first case, and a+b in the second. Also, if from a-c we take away +b or -b, the remainder, from (2), will be a-c-b, or a~c+b.

66. Hence, we have the following general rule for the subtraction of algebraic quantities.

RULE.

Change the signs of all the quantities to be subtracted into the contrary signs, or conceive them to be so changed, and then add, or connect them together, as in the several cases of addition.

EXAMPLE 1. From 18ab subtract 14ɑb.

Here, changing the sign of 14ab, it becomes -14ab, which being connected to 18ab with its proper sign, we have 18ab-14ab (18-14)ab= 4ab. Ans.

Ex. 2. From 15x2 subtract -10x2.

Changing the sign of 10x2, it becomes +10x2, which being connected to 15x2 with its proper sign, we have 15x2+10x225x2. Ans.

Ex. 3. From 24ab+7cd subtract 18ab+7cd. Changing the signs of 18ab+7cd, we have -18ab -7cd, therefore, 24ab+7cd-18ab7ed6ab.

Ex. 4. Subtract 7a-5b+3ax from 12a+10b+

13ax-3ab.

Changing the signs of

all the terms of 7a-5b

+-3ax; it becomes,

12a+10b+13ax-3ab

-7a+56-3ax

by addition, 5a+156+10ux-3ab.