Question

the expression $4(5 + w) + 6w$ is simplified in several steps below.
for each step, choose the reason that best justifies it.

step	reason
$20 + 4w + 6w$	select
$20 + 10w$	select
$10w + 20$	select

Explanation:

Step1: Identify the first step's reason

To get from \( 4(5 + w) + 6w \) to \( 20 + 4w + 6w \), we use the Distributive Property (also known as the Distributive Law), which states that \( a(b + c) = ab + ac \). Here, \( a = 4 \), \( b = 5 \), and \( c = w \), so \( 4(5 + w) = 4 \times 5 + 4 \times w = 20 + 4w \).

Step2: Identify the second step's reason

To get from \( 20 + 4w + 6w \) to \( 20 + 10w \), we combine the like terms \( 4w \) and \( 6w \). Like terms are terms with the same variable (or no variable, like constants). Combining like terms means adding their coefficients: \( 4w + 6w = (4 + 6)w = 10w \). This is the Combining Like Terms property.

Step3: Identify the third step's reason

To get from \( 20 + 10w \) to \( 10w + 20 \), we use the Commutative Property of Addition, which states that \( a + b = b + a \). Here, we swap the positions of \( 20 \) (a constant term) and \( 10w \) (a variable term), so \( 20 + 10w = 10w + 20 \).

For the first dropdown (step \( 4(5 + w) + 6w \to 20 + 4w + 6w \)):

The reason is the Distributive Property (or "Distributive Law").

For the second dropdown (step \( 20 + 4w + 6w \to 20 + 10w \)):

The reason is "Combining Like Terms" (or "Simplifying Like Terms").

For the third dropdown (step \( 20 + 10w \to 10w + 20 \)):

The reason is the Commutative Property of Addition (or "Commutative Law of Addition").

Answer:

• First step ( \( 20 + 4w + 6w \) ): Distributive Property
• Second step ( \( 20 + 10w \) ): Combining Like Terms
• Third step ( \( 10w + 20 \) ): Commutative Property of Addition