Question

an illustration of the commutative property for addition is—
\\(\frac{2a + 2b}{2} = a + b\\)
\\(2(a + b) = 2a + 2b\\)
\\(2a + 2b = 2b + 2a\\)
\\(a + (b + c) = (a + b) + c\\)

Explanation:

Step1: Recall Commutative Property of Addition

The commutative property of addition states that for any two numbers (or algebraic expressions) \( x \) and \( y \), \( x + y = y + x \). This means that the order of adding two terms does not change the sum.

Step2: Analyze Each Option

• Option 1: \(\frac{2a + 2b}{2}=a + b\) is a simplification (factoring and dividing by 2), not related to the commutative property.
• Option 2: \(2(a + b)=2a + 2b\) is the distributive property of multiplication over addition, not commutative.
• Option 3: \(2a+2b = 2b + 2a\) follows the form \(x + y=y + x\) where \(x = 2a\) and \(y=2b\), so this is the commutative property of addition.
• Option 4: \(a+(b + c)=(a + b)+c\) is the associative property of addition, which deals with the grouping of terms, not the order.

Answer:

\(2a + 2b=2b + 2a\) (the third option)