Using both the Commutative and Associative Properties, determine whether the following two expressions are equivalent: Expression #1: (3b+1)+10b+9(3𝑏+1)+10𝑏+9Expression #2: 13b+913𝑏+9 (1 point)ResponsesThe expressions are equivalent. The expressions are equivalent. The expressions are NOT equivalent.The expressions are NOT equivalent.

Answer 1

To determine whether the two expressions are equivalent, let's simplify Expression #1 and see if it matches Expression #2.

Expression #1: \[ (3b + 1) + 10b + 9(3b + 1) + 10b + 9 \]

We can start by distributing \(9\) into \((3b + 1)\): \[ = (3b + 1) + 10b + (27b + 9) + 10b + 9 \]

Now, combine like terms:

1. Combine the \(b\) terms:

   • \(3b + 10b + 27b + 10b\)
   • This equals \(3b + 10b + 27b + 10b = 50b\)

2. Combine the constant terms:

   • \(1 + 9 + 9\)
   • This equals \(1 + 9 + 9 = 19\)

Putting it all together, we have: \[ Expression #1 = 50b + 19 \]

Expression #2: \[ 13b + 9 \]

Now, we need to compare the simplified Expression #1 with Expression #2:

• Expression #1: \(50b + 19\)
• Expression #2: \(13b + 9\)

Clearly, \(50b + 19\) is not equal to \(13b + 9\).

Thus, the two expressions are not equivalent.

The final response is: The expressions are NOT equivalent.