Page:Elementary algebra (1896).djvu/33

SIMPLE BRACKETS. 15 Again, $$ a - \left ( {b - c} \right ) $$ means that from a we are to subtract the excess of b over c. If from a we take b we get $$a - b $$ but by so doing we shall have taken away c too much, and must therefore add do a - b. Thus a - (b - c) = a - b + c. In like manner, a - b - (c —d - e) = a - b - c -{- d -- e. Accordingly the following rule may be enunciated:

Rule. When an expression within brackets is preceded by the sign -, the brackets may be removed if the sign of every term wiithin the brackets be changed.

Conversely: Any part of an expression may be enclosed within brackets and the sign - prefixed, provided the sign of every term within the brackets be changed.

Thus the expression a - b--c--d - e may be written in any of the following ways, a - (--b - c - d--e), a - b - {- c - d -- e), a - b -Y c - {— d -- e).

We have now established the following results : I. Additions and subtractions may be made in any order. Thus a - b + c + d - e -f= a - c --b -- d - /- e = a - c - /+ d-{- b - e.

This is known as the Commutative Law for Addition and Subtraction.

II. The terms of an expression may be grouped in any manner.

Thus a + b-c + d-e -f = (a + l))-c-^(d - e)-f = a +(6 - c) + (c? - e)~f= a + b -{c - d)-{e +/).

This is known as the Associative Law for Addition and Subtraction.

ADDITION OF UNLIKE TERMS.

27. When two or more like terms are to be added together we have seen that they may be collected and the result