Page:Elementary algebra (1896).djvu/34

expressed as a single like term. If, however, the terms are unlike, they cannot be collected. Thus in finding the sum of two unlike quantities a and b, all that can be done is to connect them by the sign of addition and leave the result in the form $a+b$.

Also, by the rules for removing brackets, $a+\left(-b\right)=a-b$ that is, the algebraic sum of $a$ and $-b$ is written in the form $a-b$.

28. It will be observed that in Algebra the word sum is used in a wider sense than in Arithmetic. Thus, in the language of Arithmetic, $a-b$ signifies that $b$ is to be subtracted from $a$, and bears that meaning only; but in Algebra it also means the sum of the two quantities $a$ and $-b$ without any regard to the relative magnitudes of $a$ and $b$.

Ex. 1. Find the sum of $3a-5b+2c$; $2a+3b-d$;

${\begin{aligned}{\text{The sum}}&=\left(3a-5b+2c\right)+\left(2a+3b-d\right)+\left(-4a+2b\right)\\&=3a-5b+2c+2a+3b-d-4a+2b\\&=3a+2a-4a-5b+3b+2b+2c-d\\&=a+2c-d{\text{,}}\end{aligned}}$

by collecting like terms. The addition is more conveniently effected by the following rule:

Rule. Arrange the expressions in lines so that the like terms may be in the same vertical columns: then add each column, beginning with that on the left.

	The algebraic sum of the terms in the first column is $a$, that of the terms in the second column is zero. The single terms in the third and fourth columns are brought down without change.
$\quad 3a-5b+2c$
$\quad 2a+3b+2c\qquad -d$
$\,-4a+2b$
$\quad \;a\qquad \ +2c\qquad -d$

Ex. 2. Add together $-5ab+6bc-7ac$; $8ab+3ac-2ad$; $-2ab+4ac+5ad$; $bc-3ab+4ad$.

$\ -5ab+\ 6bc-7ac\qquad$
$\quad 8ab\qquad \quad \ +3ac\ -2ad$	Here we first rearrange the expressions so that like terms are in the same vertical columns, and then add up each column separately.
$\ -2ab\quad \quad \ \ \ +4ac+5ad$
$\ -3ab\ \ +\ \ bc\qquad \ \ +4ad$
$\ -2ab\ +\ \ \ 7bc\qquad +7ad$