Page 23 - Appplied Mathematics for the Petroleum and Other Industries, 5th Edition

84 PRINCIPLES OF ALGEBRA

form another unknown. These unknowns are terms. The terms of the expression
are the combinations of symbols that a sign does not separate—for example,
2a, 3ab, and 4c are the terms of the expression 2a + 3ab – 4c. The plus (+) and
minus (–) are operational signs—they tell you to add or subtract. Also, 2a, 3ab,
and 4c indicate multiplication in that 2a = 2 × a, 3ab = 3 × a × b, and 4c = 4 × c.
Thus, algebraic expressions may be added, subtracted, multiplied, and divided.

Example Problem: If a = 6, b = 4, and c = 3, find the value of 2a + 3b + 5c.
Solution: Substitute numerical values for letters:
(2 × 6) + (3 × 4) + (5 × 3) = 12 + 12 + 15 = 39.

Grouping Terms
Another important aspect of algebraic expressions is that the terms within them
may be grouped. Grouping terms indicates the order in which these operations
should be carried out. Parentheses (), brackets [], and braces {} may be used
for grouping terms. These symbols indicate that you should keep certain terms
together; they also indicate the order in which you should perform the various
operations in the expression. These grouping symbols must be removed before
addition or subtraction can be performed. Removing parentheses, brackets, and
braces means performing the operation indicated inside them before working
the rest of the expression. A few rules govern grouping symbols.
Rule 1. When an expression within the parentheses is preceded by the plus sign,
the parentheses are removed without changing the signs within the parentheses.
Rule 2. When an expression within the parentheses is preceded by the minus
sign, the parentheses are removed by changing the signs within the parentheses.
Rule 3. A number or symbol before the parentheses indicates that each term
within the parentheses is to be multiplied by that number or symbol.

Example Problem: Simplify the expression, 3(a – b) + 4(b + c).

Solution: First remove the parentheses. Perform this calculation by under-
standing exactly what the expression states. In this case, the expression says
to multiply a – b by 3, then add the result to 4 times b + c. So, to remove
the parentheses:
3 × a and 3 × b = 3a and 3b; in the same way, 4 × b = 4b and 4 × c = 4c. The
result is
3a – 3b + 4b + 4c.
Then, combine like terms by performing the subtraction and addition:
3a + b + 4c.
Petroleum Extension-The University of Texas at Austin
Notice that only one part of the entire expression has like terms, which
are +4b and –3b. Subtracting 3b from 4b leaves 1b, or simply b. Like terms are
terms whose letters or symbols are the same. Just as you cannot add or subtract
apples from oranges—they are unlike terms—you cannot add or subtract a from
b, b from c, and so on. You can, of course, add or subtract like terms—apples and
apples or oranges and oranges, as it were. In the case of the previous problem,
subtract 3b from 4b to get b.