The expression you have uploaded is:

$\frac{\frac{1}{2} xz + \frac{1}{4} yz - \frac{2}{3} z - 3xz + 2z}{4 - \frac{4}{3} z}$

Let's simplify this step by step:

1. Simplify the numerator:

   The numerator is:

   $\frac{1}{2} xz + \frac{1}{4} yz - \frac{2}{3} z - 3xz + 2z$

   • Combine the terms involving $xz$: $\frac{1}{2} xz - 3xz = \left( \frac{1}{2} - 3 \right)xz = \left( \frac{1}{2} - \frac{6}{2} \right)xz = -\frac{5}{2} xz$

   • Combine the terms involving $z$: $\frac{2}{3} z + 2z = \left( \frac{2}{3} + \frac{6}{3} \right)z = \frac{8}{3} z$

   So, the numerator simplifies to: $-\frac{5}{2} xz + \frac{1}{4} yz + \frac{8}{3} z$

2. Simplify the denominator:

   The denominator is: $4 - \frac{4}{3} z$

   Leave this as is for now.

3. Final expression:

   The simplified version of the expression is:

   $\frac{-\frac{5}{2} xz + \frac{1}{4} yz + \frac{8}{3} z}{4 - \frac{4}{3} z}$

Let me know if you need further details or if you'd like to explore the expression in a different way!

5 Related Questions:

1. How can we factor the simplified numerator further?
2. Can we solve this expression for $z$ if specific values for $x$ and $y$ are provided?
3. What happens when $z = 0$? How does the expression behave?
4. Is there any way to express this result in terms of partial fractions?
5. How would this expression change if the constants (like 1/2, 1/4, etc.) were different?

Tip:

When simplifying expressions, always group like terms first. This makes the process much easier and clearer!