https://file.aiquickdraw.com/mathbot/file/17261128390256hqpiorr.jpg

Let's analyze both expressions in detail to see if they are equivalent.

### First Expression:
\[
\frac{1}{2}x^2yz - \frac{1}{4}x^2yz
\]
This expression can be simplified by factoring out \(x^2yz\):
\[
x^2yz \left( \frac{1}{2} - \frac{1}{4} \right)
\]
Now, simplify the fraction inside the parentheses:
\[
\frac{1}{2} - \frac{1}{4} = \frac{2}{4} - \frac{1}{4} = \frac{1}{4}
\]
Thus, the first expression simplifies to:
\[
\frac{1}{4}x^2yz
\]

### Second Expression:
\[
-\frac{1}{2}x^2yz
\]
This is already in its simplest form.

### Comparison:
The simplified first expression is \(\frac{1}{4}x^2yz\), while the second expression is \(-\frac{1}{2}x^2yz\). These two are **not equivalent**, because \(\frac{1}{4} \neq -\frac{1}{2}\).

Therefore, the pair of expressions **is not equivalent**.

Would you like further details or have any questions? Here are some related questions to consider:

1. How do you simplify expressions with fractions?
2. What does factoring out a common term mean?
3. How can you compare two algebraic expressions efficiently?
4. What are some other examples of equivalent and non-equivalent expressions?
5. How does the distributive property work in simplifying expressions?

**Tip:** Always simplify both expressions as much as possible before comparing them!

Tell whether this pair of expressions is equivalent: \(\frac{1}{2}x^2yz - \frac{1}{4}x^2yz\) and \(-\frac{1}{2}x^2yz\).

This problem helps students determine if two algebraic expressions, involving factoring and simplification of fractions, are equivalent. Concepts such as factoring out common terms and applying the distributive property are covered. Suitable for Grades 8-10.

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