Question

which expressions are equivalent to $-6z + (-2.5) + 1.5z + 7y - 3.5$?
select all that apply.
a $ -6 + 7y + (-4.5z)$
b $ -4.5z + 7y - 6$
c $ 7y + (-4.5z) - 2.5$
d $ 7y - 6 - 4.5z$
e $ -6 + 7y + 4.5z$

Explanation:

Step1: Simplify the original expression

First, simplify the constants and the \( z \)-terms in the original expression \( -6z + (-2.5) + 1.5z + 7y - 3.5 \). Combine the constant terms: \( -2.5 - 3.5=-6 \). Combine the \( z \)-terms: \( -6z + 1.5z=-4.5z \). So the original expression simplifies to \( 7y - 4.5z - 6 \), which can be rewritten in different orders.

Step2: Analyze Option A

Option A: \( -6 + 7y + (-4.5z) \). Rearranging the terms of our simplified expression \( 7y - 4.5z - 6 \) (using commutative property of addition: \( a + b + c = c + b + a \)), we get \( -6 + 7y - 4.5z \), and \( -4.5z \) is the same as \( +(-4.5z) \). So this is equivalent.

Step3: Analyze Option B

Option B: \( -4.5z + 7y - 6 \). Rearranging the terms of our simplified expression \( 7y - 4.5z - 6 \) (using commutative property of addition: \( a + b + c = b + a + c \)), we get \( -4.5z + 7y - 6 \). So this is equivalent.

Step4: Analyze Option C

Option C: \( 7y + (-4.5z) - 2.5 \). The constant term here is \( -2.5 \), but in our simplified expression the constant term is \( -6 \) (since \( -2.5 - 3.5=-6 \)). So this is not equivalent.

Step5: Analyze Option D

Option D: \( 7y - 6 - 4.5z \). Rearranging the terms of our simplified expression \( 7y - 4.5z - 6 \) (using commutative property of addition: \( a + b + c = a + c + b \)), we get \( 7y - 6 - 4.5z \). So this is equivalent.

Step6: Analyze Option E

Option E: \( -6 + 7y + 4.5z \). Our simplified expression has \( -4.5z \), not \( +4.5z \). So this is not equivalent.

Answer:

A. \( -6 + 7y + (-4.5z) \), B. \( -4.5z + 7y - 6 \), D. \( 7y - 6 - 4.5z \)