Simplifying by Combining Like Terms
Simplifying by Combining Like Terms teaches students to reduce algebraic expressions by adding or subtracting the coefficients of terms that share the same variable and exponent. From OpenStax Prealgebra 2E, the key rule: capture the sign immediately to the left of each term — it belongs to that term. Simplifying 7b + 5 − 3b + 2 groups b-terms (7b − 3b = 4b) and constants (5 + 2 = 7) to get 4b + 7. In 4x − 8y − x + 3y, the x-terms give 3x and the y-terms give −5y. Never change the variable or exponent when combining.
Key Concepts
Property To simplify an expression means to combine all possible like terms into a single term. 1. Group the like terms together (mentally or physically). 2. Add or subtract their coefficients . 3. Never change the variable or the exponent when adding or subtracting! Crucial Rule: Always "capture" the sign ($+$ or $ $) immediately to the left of a term. The sign belongs to that term. Treating subtraction as "adding a negative" prevents most errors.
Examples Basic Grouping: Simplify $7b + 5 3b + 2$. Combine the $b$'s: $7b 3b = 4b$. Combine the constants: $5 + 2 = 7$. Final Answer: $4b + 7$. Capturing the Sign: Simplify $4x 8y x + 3y$. The terms are: $4x$, $ 8y$, $ 1x$, $+3y$. Combine the $x$'s: $4x 1x = 3x$. Combine the $y$'s: $ 8y + 3y = 5y$. Final Answer: $3x 5y$.
Explanation The most common mistake students make is dropping or confusing negative signs. When you see $5x 8x$, do not think of it as "five minus eight." Think of it as combining a positive $5$ and a negative $8$, which results in a negative $3$ ($ 3x$). Circling or underlining like terms (along with the $+$ or $ $ sign right in front of them) is the best way to keep your math clean and accurate.
Common Questions
What are like terms?
Like terms have the same variable raised to the same power. 3x and 7x are like terms; 3x and 3x² are not.
How do you simplify 7b + 5 − 3b + 2?
Group b-terms: 7b − 3b = 4b. Group constants: 5 + 2 = 7. Result: 4b + 7.
How do you simplify 4x − 8y − x + 3y?
The terms are 4x, −8y, −x, +3y. Combine x-terms: 4x − x = 3x. Combine y-terms: −8y + 3y = −5y. Result: 3x − 5y.
What does 'capturing the sign' mean?
The sign (+ or −) immediately before a term belongs to that term. −3b means negative 3b, so you must include the negative when combining.
Can you combine 4x and 4x²?
No. The exponents differ, so 4x and 4x² are unlike terms and cannot be combined.
What changes when you combine like terms?
Only the coefficient changes. The variable and exponent stay the same: 5x + 3x = 8x, not 8x².