Page:Elementary algebra (1896).djvu/34

16 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 $$3a - 5b + 2c$$; $$2a + 3b - d$$;

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

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.

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