Algebra I, Article 53. Adding Similar Quantities with Like Signs

In this lesson, we will begin the study of algebraic addition. To complete the objectives for this lesson, complete the following tasks:

  1. Read the lesson from beginning to end for familiarity.
  2. Study the lesson for mastery.
  3. Memorize the Rule to Add Similar Quantities with Like Signs
  4. Complete the lesson assessment.


53. Addition, in Algebra, is the process of finding the simplest expression for the sum of two or more algebraic quantities.

1. James has 3 pockets, each containing apples: in the first he has 3 apples, in the second 4, and in third 5.

In order to find how many apples he has, suppose he proceeds to find their sum in the following manner:  3 apples + 4 apples + 5 apples = 12 apples. But, instead of writing the word apples, suppose he should use the letter a, thus:  3a + 4a + 5a = 12a. It is evident that the sum of 3 times a, 4 times a, and 5 times a, is 12 times a, or 12a, whatever a may represent.

2. In the same manner the sum of -3a, -4a, and -5a would be -12a. For -3a and -4a make -7a, plus -5a make -12a.

Rule to Add Similar Quantities with Like Signs
1. Add together the coefficients of the several quantities; 
2. To their sum prefix the common sign, and annex the common, letter or letters.

Note:  When a quantity has no coefficient, 1 is understood; thus, a=1a.


Use the rule above to find the sums.

1. Add: 3a, 4a, 5a, 12a.

2. Add: -3a, -4a, -5a, -12a.

3. Add: 3a, 2a, a, 5a.  

4. Add: -6xy, -xy, -4xy, -3xy.

5. Add: 2a2, 3a2, 5a2, 7a2.

6. Add: -3a2b, -4a2b, -5a2b, -2a2b.

7. What is the sum of 3b, 5b, 7b and 9b?

8. What is the sum of 2ab, 5ab, 8ab and 11ab?

9. What is the sum of abc, 3abc, 7abc and 12abc?

10. What is the sum of -by, -2by, -5by, and -8by?

11. Add: 3ay+7, ay+8, 2ay+4, and 5ay+6.

12. Add: 8x-4y, 5x-3y, 7x-6y, 6x-2y.

13. Add: 3a2-2ax, 5a2-3ax, 7a2-5ax, and 4a2-4ax.

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