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Subtract the less coefficient from the greater, to the remainder prefix the sign of the greater, and annex their common letters or quantities.

* In example 3, the coefficients of the two quantities, viz. +2c and -2c, are equal to each other, therefore they destroy one another, and so their sum makes o, or *, which is frequently used, in algebra, to signify a vacant place.

NOTE. When many like quantities are to be added together, whereof some are affirmative and others negative; reduce them first to two terms, by adding all the affirmative quantities together, and all the negative ones; and then add the two terms according to the rule. Thus,

11. Add 4a2+7a2-3a2+12a2-8a2+a2-5a2 together.

First, 4a+7a2+12a2 +a2=24a", the sum of the affirmative quantities,

And -3a2-8a5a2-16a2, the sum of the negative.

Then 24a-16a2=8a", the sum of the whole.

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12. Add 5ax2-4ax2+10ax2-8ax2-6ax2 together.

First, 5ax2+10ax2 = 15ax2,

And-4ax2-8ax2-6x2-18ax2;

Therefore the sum of these quantities is +15ax2-18 x2=-zax2.

CASE I11.

When the quantities are unlike.

RULE.

Set them down in a line, with their signs and coefficients prefixed.

Sum za+2b-8xy+5y+2a2-b2-9

Sum 15a2-763+3ab+6c2-5d+xy-2y3+4*

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* Here the first column is composed of like quantities, which are added together by case 1. The terms -963 and +963 destroy one another; and the sum of -1263 and +563 is -763,

by case 2. The sum of +7 ab and -4 ab is +3ab. In like manner, + 10 and -6 together make +4; and the rest of the terms being unlike, they are set down with their respective signs and coefficients prefixed, conformably to case 3.

Change each + into-, and each - into +, in the subtrahend, or suppose them to be thus changed; then proceed as in addition, and the sum will be the true remainder.

EXAMPLES.

* This rule is founded on the consideration, that addition and subtraction are opposite to each other in their nature and operation, as are the signs + and -, by which they are expressed and represented. And since to unite a negative with a positive quantity of the same kind has the effect of diminishing it, or subducting an equal positive quantity from it; therefore to subtract a positive, which is the opposite of uniting or adding, is to add the equal negative quantity. In like manner, to subtract a negative quantity is the same in effect, as to add or unite an equal positive quantity. So that, by changing the sign of a quantity from + *to-, or from - to +, its nature is changed from a subductive to an additive quantity; and any quantity is in effect subtracted by barely changing its sign.

* The ten foregoing examples of simple quantities being obvious, we pass by them; but shall illustrate the eleventh example, in order to the ready understanding of those, which follow. In the eleventh example, the compound quantity 2ax2+4 being tąken from the simple quantity 5ax, the remainder is 3ax2-4, and it is plain, that the more there is taken from any number or quantity, the less will be left ; and the less there is taken, the more will be left. Now, if only 2ax were taken from 5ax2, the remainder would be gax2; and consequently, if zax2+4, which is greater than 2ax by 4, be taken from 5ax2, the remainder will be less than 3ax by 4, that is, there will remain 3ax2-4, as above. For by changing the sign of the quantity 2ax2+4, and adding it to 5ax2, the sum is 5ax2-2ax2-4; but here the term -2ax destroys so much of 5ax as is equal to itself, and 30 5ax2-2ax2-4 becomes equal to 3ax2-4, by the general rule for subtraction.

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