You know already in the previous topic (The Algebraic Terms) that only like terms can be added. For example, 2a + 3a = 5a. Terms containing the same letter are added just like named numbers in arithmetic, such as 2 apples + 3 apples = 5 apples; 2 cents + 3 cents = 5 cents. But if you have to add 2a + 3b, the addition cannot be performed. It would be like adding 2 apples + 3 hours. These quantities do not go together and the addition does not make sense. When all terms in an expression are like terms, they are added according to the following rule:
Rule: Coefficient of like terms, together with their signs, are added. The algebraic sum of the coefficients is the coefficient of the result; write it with the same base letter as in each term. For example, 7x + 3x + 2x = 12x. The expression represents 7, 3, and 2 like terms. If you add them, you will have 12 of the same items. The “items” are here indicated as “x,” and the result is 12x.
Study the following addition of like terms. When you go over these problems a second time, cover the answers given and try to determine the correct answers by yourself.
5a + 8a + a = 14a
18ab + ab + 7ab = 26ab
x3+ 2x3 + 3x3 = 6x3
7x2y3z + x2y3z + 2x2y3z = 10x2y3z
3mn + 2mn + 10mn = 15mn
5rts + 8str + trs = 14rts
The preceding examples consist of terms with positive coefficients. The method of addition is, therefore, the same as for the arithmetical addition of named numbers. But algebraic terms may also be negative; that is, they may have negative coefficients. In adding positive and negative terms, the rules for algebraic addition are applied. Let’s try to solve the following problems: -13bc + 5bc =? First find the algebraic sum of the coefficients, or -13 + 5 = -8. It is not necessary to use parentheses; in this problem it is clear that +5 has to be added to -13. Following the rule for the algebraic addition of numbers with opposite signs, subtract the coefficients. The result will have the sign of the coefficient with the higher absolute value. Next to the resulting coefficient write the letters bc, which are common to both numbers. The result is -13bc + 5bc = -8bc
When an expression has several like terms, find their algebraic sum as follows: First, add all the positive terms together; then add all the negative terms together; finally, find the algebraic sum of these two results. For example, 17mn – 8mn + 6mn – 11mn + mn =?
These are all like terms because they contain the same letters mn. Add the three positive terms first: 17mn + 6mn + mn = 24mn. You must add in your mind the coefficients 17 + 6 + 1 = 24, and then attach mn to 24. Remember that the terms mn has a coefficient + 1. Next add together the negative terms, or -8mn – 11mn = 19mn. Since both terms are negative, the result is negative. Now, find the algebraic sum: 24mn – 19mn = 5mn.
This problem looks like subtraction, and you do actually subtract the coefficients (24-19), but the result is called the algebraic sum because you have added algebraically the term +24 and the term -19mn. Because these terms have different signs, the algebraic sum equals the difference between the absolute values and takes the sign of the term with the higher absolute value.