Question

the expression $5(2 + u) + 9u$ is simplified in several steps below.
for each step, choose the reason that best justifies it.

step	reason
$10 + 5u + 9u$	select
$10 + 14u$	select
$14u + 10$	select

Explanation:

for Step 1 (10 + 5u + 9u):

Step1: Identify the property used

To get from \( 5(2 + u) + 9u \) to \( 10 + 5u + 9u \), we use the Distributive Property (also known as the Distributive Law). The distributive property states that \( a(b + c) = ab + ac \). Here, \( a = 5 \), \( b = 2 \), and \( c = u \), so \( 5(2 + u) = 5 \times 2 + 5 \times u = 10 + 5u \). Then we still have the \( + 9u \) from the original expression, so we get \( 10 + 5u + 9u \).

for Step 2 (10 + 14u):

Step1: Identify the property used

To get from \( 10 + 5u + 9u \) to \( 10 + 14u \), we use the Combining Like Terms (or "Combine Like Terms") property. Like terms are terms with the same variable raised to the same power. Here, \( 5u \) and \( 9u \) are like terms (both have the variable \( u \) with an exponent of 1). To combine them, we add their coefficients: \( 5u + 9u = (5 + 9)u = 14u \). So we replace \( 5u + 9u \) with \( 14u \) to get \( 10 + 14u \).

for Step 3 (14u + 10):

Step1: Identify the property used

To get from \( 10 + 14u \) to \( 14u + 10 \), we use the Commutative Property of Addition. The commutative property of addition states that \( a + b = b + a \), which means we can swap the order of the terms being added. Here, \( a = 10 \) and \( b = 14u \), so \( 10 + 14u = 14u + 10 \).

Answer:

s for Each Step:

• For \( 10 + 5u + 9u \): Reason is "Distributive Property" (or "Distributive Law")
• For \( 10 + 14u \): Reason is "Combine Like Terms" (or "Combining Like Terms")
• For \( 14u + 10 \): Reason is "Commutative Property of Addition"