Page:Elementary algebra (1896).djvu/29

22. Addition is the process of finding in simplest form the algebraic sum of any number of quantities.

23. Like terms, or similar terms, do not differ, or differ only in their numerical coefficients. Other terms are called unlike, or dissimilar. Thus $3a$, $7a$; $5a^{2}b$, $2a^{2}b$; $3a^{3}b^{2}$, $-4a^{3}b^{2}$ are pairs of like terms;and $4a$, $3b$; $7a^{2}$, $9a^{2}b$ are pairs of unlike terms.

ADDITION OF LIKE TERMS.

Rule I. The sum of a number of like terms is a like term.

Rule II. If all the terms are positive, add the coefficients.

Ex. Find the value of $8a+5a$.

Here we have to increase 8 like things by 5 like things of the same kind, and the aggregate is 13 of such things;

for instance,

$8{\text{ lbs.}}+5{\text{ lbs.}}=13{\text{ lbs.}}$.

Hence also,

$8a+5a=13a$.

Similarly,

$8a+5a+a+2a+6a=22a$.

Rule III. If all the terms are negative, add the coefficients numerically and prefix the minus sign to the sum.

Ex. To find the sum of $-3x$, $-5x$, $-7x$, $-x$.

Here the word sum indicates the aggregate of 4 subtractive quantities of like character. In other words, we have to take away successively $3,5,7,1$ like things, and the result is the same as taking away $3+5+7+1$ such things in the aggregate.

Thus the sum of $-3x,-5x,-7x,-x$, is $-16x$.

Rule IV. If the terms are not all of the same sign, add together separately the coefficients of allthe positive terms and the coefficients of all the negative terms; the difference of these two results, preceded by the sign of the greater, will give the coefficient of the sum required.

Ex. 1. The sum of $17x$ and $-8x$ is $9x$, for the difference of $17$ and $8$ is $9$, and the greater is positive.

Ex. 2. To find the sum of $8a,-9a,-a,3a,4a,-11a,a$.

The sum of the coefficients of the positive terms is 16. The sum of the coefficients of the negative terms is 21.

The difference of these is 5, and the sign of the greater is negative; hence the required sum is $-5a$.