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!