Equivalent Linear Expressions

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Concept Review

Like Terms: Finding Your Mathematical Family

Imagine you're organizing a messy closet full of clothes. You'd naturally group all the shirts together, all the pants together, and all the socks together. In algebra, we do something remarkably similar with like terms — we identify which parts of an expression belong in the same "mathematical family."

A like term is any term in an algebraic expression that has exactly the same variable part. Think of the variable part as the "family name" — terms can only be combined if they share the same family identity.

Spotting the Family Resemblance

Let's examine this expression: 3x + 7 + 5x - 2 + x

To identify like terms, we look at the variable part of each term:

3x→ variable part is "x"
7→ no variable (constant term)
5x→ variable part is "x"
-2→ no variable (constant term)
x→ variable part is "x"

The x-terms (3x, 5x, and x) are like terms because they share the same variable part. The constant terms (7 and -2) are like terms because they both have no variables.

🔑 Key Insight

The coefficient (the number in front) doesn't matter for identifying like terms — only the variable part matters. So 2x and -9x are like terms, but 2x and 2y are not like terms, even though they have the same coefficient!

The Variable Part Detective Work

Consider the expression: 4a + 2b - 7a + 9 + 3b - 1

a-family

4a, -7a

b-family

2b, 3b

constant family

9, -1

Key Takeaway: Just like organizing that messy closet, identifying like terms is about recognizing what belongs together. By grouping terms with identical variable parts, you're preparing to combine them and simplify expressions — turning algebraic chaos into organized, manageable mathematics.

Sample questions

1. In the expression 3x + 5y - 2x + 7, which terms are like terms?

✓ 3x and -2x

○ 3x and 5y

○ 5y and 7

○ -2x and 7

Answer: 3x and -2x — Like terms have the same variable raised to the same power. 3x and -2x both have x.

2. Identify the like terms in: 4a - 3b + 2a + 5b - 8.

○ 4a and -3b; 2a and 5b

○ 4a and 5b; 2a and -3b

✓ 4a and 2a; -3b and 5b

○ 4a and -8; 2a and -8

Answer: 4a and 2a; -3b and 5b — Terms with a: 4a and 2a. Terms with b: -3b and 5b.

3. Which term is NOT like the others in the expression 6x - 2y + 3x + 4y?

○ 6x and 3x are like terms, -2y and 4y are like terms, so all are paired

○ -2y and 4y are like terms

○ 6x and -2y have no like terms with each other

✓ All terms have like terms in the expression

Answer: All terms have like terms in the expression — Every term has a matching like term: 6x with 3x, -2y with 4y.

Skills in this topic

Identify like terms in a linear expression
Combine like terms with positive and negative coefficients
Identify equivalent linear expressions
Use properties of operations to simplify linear expressions
Evaluate equivalent expressions to prove they yield the same result

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