You have the equation \(50z + 60x + 30y = 5290\) with the condition \(z > y > x\). To solve for the possible values of \(z\), \(y\), and \(x\), we can proceed by finding integer values that satisfy both the equation and the inequality condition.

### Step-by-Step Solution

1. **Simplify the equation**:  
   Divide the entire equation by 10 to make it simpler:
   \[
   5z + 6x + 3y = 529
   \]

2. **Express one variable in terms of the others**:  
   Let's express \(z\) in terms of \(x\) and \(y\):
   \[
   z = \frac{529 - 6x - 3y}{5}
   \]
   For \(z\) to be an integer, \(529 - 6x - 3y\) must be divisible by 5.

3. **Check divisibility**:  
   \[
   529 - 6x - 3y \equiv 0 \ (\text{mod} \ 5)
   \]
   Simplify modulo 5:
   \[
   529 \equiv 4 \ (\text{mod} \ 5)
   \]
   \[
   6x \equiv x \ (\text{mod} \ 5)
   \]
   \[
   3y \equiv 3y \ (\text{mod} \ 5)
   \]
   So:
   \[
   4 - x - 3y \equiv 0 \ (\text{mod} \ 5)
   \]
   \[
   x + 3y \equiv 4 \ (\text{mod} \ 5)
   \]

   Now, we test values of \(y\) (since \(y > x\), and both must be integers) and find corresponding \(x\) and \(z\).

### Solution by Testing
Let's test values for \(y\) and solve for \(x\) and \(z\):

- **For \(y = 3\)**:  
  \[
  x + 9 \equiv 4 \ (\text{mod} \ 5) \Rightarrow x \equiv 0 \ (\text{mod} \ 5)
  \]
  So, \(x = 0, 5, 10, \dots\), but \(x < y = 3\), so no valid \(x\).

- **For \(y = 4\)**:  
  \[
  x + 12 \equiv 4 \ (\text{mod} \ 5) \Rightarrow x \equiv 2 \ (\text{mod} \ 5)
  \]
  So, \(x = 2\) is valid.

  Substitute \(x = 2\) and \(y = 4\) into the simplified equation:
  \[
  5z + 6(2) + 3(4) = 529
  \]
  \[
  5z + 12 + 12 = 529
  \]
  \[
  5z = 505 \Rightarrow z = 101
  \]

Therefore, one solution is \(z = 101\), \(y = 4\), \(x = 2\).

### Final Answer
The values satisfying the given equation and conditions are:
\[
z = 101, \ y = 4, \ x = 2
\]

Do you want more details or have any questions about this solution?

---

1. How can we solve such problems when the coefficients are different?
2. What if we have non-integer solutions?
3. What strategies are useful for solving inequalities with multiple variables?
4. How do modulus operations help in solving equations?
5. Can this problem be solved using a graph?

**Tip:** Always simplify equations by dividing by common factors to reduce complexity.

Learn how to solve the linear equation 50z + 60x + 30y = 5290 with the condition z > y > x. This problem involves linear equations, inequalities, and modular arithmetic. Suitable for advanced high school students.

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