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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 --26, 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 quaritities together, and all the negative ones ; and then add the two terms according to the rule. Thus,
11. Add 40* +7a*---3a* + 122*--8a* ta'-5a* together.
First, 4a * ta*+12a* ta'=24a2, the sum of the affirmative quantities,
And —30-80-50*=16a*, the sum of the negative.
Then 24a-160= 8a, the sum of the whole.
Therefore the sum of these quantities is +-150x180 x=-3ax.
When the quantities are unlike.
Set them down in a line, with their signs and coefficients prefixed.
Sum 152*—763+3V ab +6c*-50+xy—2y3+4*
* Here the first column is composed of like quantities, which are added together by case 1. The terms -963 and +96% destroy one another ; and the sum of —1263 and +563 is -763, by case 2. The sum of +7V ab and --4V ab is +3v ab. 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 ti in the subtrahend, or suppose them to be thus changed ; then proceed as in addition, and the sum will be the true remainder.
* 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 rep- . résented. 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 t, its nature is changed from a subductive to an additive quantity; and any 'quantity is in effect subtractrd by barely changing its sign.
* The ten foregoing examples of simple quantities being obvi. ous, 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 2ax: +4 being taken from the simple quantity 5ax”, the remainder is zax?-4, and it is plain, that the more there is taken from
quan • tity, the less will be left ; and the less there is taken, the more will be left. Now, if only zax2 were taken from 5axạ, the remainder would be zax? ; and consequently, if zax: + 4, which is greater than zaxa by 4, be taken from 5ax”, the remainder will be less than zax* by 4, that is, there will remain 3ax?—-4, as above. For by changing the sign of the quantity 20x2+4, and adding it to 50x?, the sum is 5ax?-2ax2-4 ; but here the
—2ax? destroys so much of 5ax? as is equal to itself, and so sax: -2ax?- 4 becomes equal to zax'-4, by the general rule for subtraction,