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We are given that points \( X \), \( Y \), and \( Z \) lie on a number line at positions \( -2 \), \( 0 \), and \( x_3 \), respectively. The weights of these points are as follows:

- Point \( X \) has twice the weight of point \( Y \).
- Point \( Z \) has three times the weight of point \( Y \).
- The weighted average of the three points is located at \( \frac{4}{3} \).

To find the location of point \( Z \), we use the formula for a weighted average:

\[
\text{Weighted average} = \frac{w_X x_X + w_Y x_Y + w_Z x_Z}{w_X + w_Y + w_Z}
\]

Where:
- \( w_X \), \( w_Y \), and \( w_Z \) are the weights of points \( X \), \( Y \), and \( Z \), respectively.
- \( x_X \), \( x_Y \), and \( x_Z \) are the positions of points \( X \), \( Y \), and \( Z \).

### Step-by-Step Solution:

1. **Assign weights**:
   - Let the weight of point \( Y \) be \( w_Y = 1 \).
   - The weight of point \( X \) is \( w_X = 2 \), since it weighs twice as much as \( Y \).
   - The weight of point \( Z \) is \( w_Z = 3 \), since it weighs three times as much as \( Y \).

2. **Substitute positions and weights**:
   - The position of \( X \) is \( x_X = -2 \).
   - The position of \( Y \) is \( x_Y = 0 \).
   - The position of \( Z \) is \( x_Z \) (which we need to find).
   - The weighted average is given as \( \frac{4}{3} \).

Now, substitute all these into the weighted average formula:

\[
\frac{4}{3} = \frac{2(-2) + 1(0) + 3(x_Z)}{2 + 1 + 3}
\]

3. **Simplify the equation**:
   \[
   \frac{4}{3} = \frac{-4 + 0 + 3x_Z}{6}
   \]
   \[
   \frac{4}{3} = \frac{-4 + 3x_Z}{6}
   \]

4. **Multiply both sides by 6 to eliminate the denominator**:
   \[
   8 = -4 + 3x_Z
   \]

5. **Solve for \( x_Z \)**:
   \[
   8 + 4 = 3x_Z
   \]
   \[
   12 = 3x_Z
   \]
   \[
   x_Z = 4
   \]

### Final Answer:
The location of point \( Z \) is \( x_Z = 4 \).

---

Do you want any further clarification on this solution? 

Here are 5 related questions to expand your understanding:
1. What is the general formula for a weighted average in higher dimensions?
2. How would the solution change if point \( Z \) had a different weight, say twice the weight of point \( Y \)?
3. How does the concept of center of mass relate to weighted averages in mathematics?
4. If the points were on a 2D plane, how would the weighted average be calculated?
5. How can we apply weighted averages in probability and statistics?

**Tip:** Always check the sum of weights when dealing with weighted averages to ensure accuracy in calculations.

Points X, Y, and Z lie on a number line at -2, 0, and x3, respectively. Point X weighs twice as much as point Y, and point Z weighs three times as much as point Y. The weighted average is located at 4/3. Explain how to find the location of point Z.

Learn how to find the location of point Z on a number line using weighted averages. Given that points X, Y, and Z lie at -2, 0, and an unknown position, and their weights differ, discover the step-by-step method to determine Z's location. Ideal for high school students studying algebra.

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