Using Properties to Group Like Terms
Using Properties to Group Like Terms shows how Commutative and Associative Properties make simplifying algebraic expressions more manageable. From OpenStax Prealgebra 2E, students rearrange terms so like terms are adjacent before combining. The expression 3x + 7 + 4x + 5 is reordered to 3x + 4x + 7 + 5 = 7x + 12 by applying the Commutative Property. Similarly, 18p + 6q + (−15p) + 5q reorders to group p-terms and q-terms: 3p + 11q. This strategic grouping reduces errors and streamlines multi-term simplification.
Key Concepts
Property When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first. We can rearrange an expression so the like terms are together. For example, we simplify $3x + 7 + 4x + 5$ by rewriting it as $3x + 4x + 7 + 5$ and then combining like terms to get $7x + 12$. We were using the Commutative Property of Addition.
Examples To simplify $18p + 6q + ( 15p) + 5q$, reorder the terms: $18p + ( 15p) + 6q + 5q$. This combines to $3p + 11q$.
To simplify $\frac{7}{15} \cdot \frac{8}{23} \cdot \frac{15}{7}$, reorder the factors to group reciprocals: $\frac{7}{15} \cdot \frac{15}{7} \cdot \frac{8}{23}$. This becomes $1 \cdot \frac{8}{23} = \frac{8}{23}$.
Common Questions
How do you use properties to simplify 3x + 7 + 4x + 5?
Apply the Commutative Property to reorder: 3x + 4x + 7 + 5. Combine like terms: 7x + 12.
How do you simplify 18p + 6q + (−15p) + 5q?
Reorder to group like terms: 18p + (−15p) + 6q + 5q = 3p + 11q.
Why rearrange terms before combining?
Grouping like terms together visually reduces mistakes when adding coefficients, especially in long expressions with many variables.
How do you simplify (7/15) · (8/23) · (15/7)?
Reorder to group reciprocals: (7/15) · (15/7) · (8/23) = 1 · (8/23) = 8/23.
Which properties justify rearranging terms?
The Commutative Property (changing order) and the Associative Property (changing grouping) both support rearranging terms in addition and multiplication.
What are like terms?
Like terms have the same variable raised to the same power — for example, 3x and 4x are like terms, but 3x and 3x² are not.