https://file.aiquickdraw.com/mathbot/file/1727056752477quy0o3jo.jpg

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!

Simplify the given algebraic expression involving variables x, y, and z.

Explore how to simplify complex algebraic expressions with variables x, y, and z. This problem demonstrates combining like terms, working with fractional coefficients, and handling variables in algebraic expressions. Suitable for high school students learning advanced algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation