I. ` (x+y)^(3) = 1331" "` II. ` x - y + z = 0`
III. ` xy = 28`

Options

A. `x lt y = z `
B. ` x le y lt z `
C. ` x lt y gt z `
D. ` x = y = z or if none of the above relationship is established

Explanation

To solve the given equations step by step, let's start with the first equation and work our way through the others.

### Step 1: Solve the first equation
We have the equation:
\[ (x + y)^3 = 1331 \]
To find \(x + y\), we take the cube root of both sides:
\[ x + y = \sqrt[3]{1331} \]
Calculating the cube root:
\[ x + y = 11 \]

### Step 2: Use the second equation
The second equation is:
\[ x - y + z = 0 \]
We can rearrange this to express \(z\) in terms of \(x\) and \(y\):
\[ z = y - x \]

### Step 3: Use the third equation
The third equation is:
\[ xy = 28 \]
Now we have two equations involving \(x\) and \(y\):
1. \(x + y = 11\)
2. \(xy = 28\)

### Step 4: Substitute \(y\) in terms of \(x\)
From the first equation, we can express \(y\) in terms of \(x\):
\[ y = 11 - x \]

### Step 5: Substitute into the third equation
Substituting \(y\) into the third equation:
\[ x(11 - x) = 28 \]
Expanding this gives:
\[ 11x - x^2 = 28 \]
Rearranging this into standard quadratic form:
\[ x^2 - 11x + 28 = 0 \]

### Step 6: Factor the quadratic equation
To factor \(x^2 - 11x + 28\), we look for two numbers that multiply to \(28\) and add to \(-11\). The factors are \(-4\) and \(-7\):
\[ (x - 4)(x - 7) = 0 \]

### Step 7: Solve for \(x\)
Setting each factor to zero gives us:
\[ x - 4 = 0 \quad \Rightarrow \quad x = 4 \]
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]

### Step 8: Find corresponding \(y\) values
Using \(y = 11 - x\):
- If \(x = 4\), then \(y = 11 - 4 = 7\).
- If \(x = 7\), then \(y = 11 - 7 = 4\).

Thus, we have two pairs: \((x, y) = (4, 7)\) and \((7, 4)\).

### Step 9: Calculate \(z\)
Using \(z = y - x\):
- For \((4, 7)\): \(z = 7 - 4 = 3\).
- For \((7, 4)\): \(z = 4 - 7 = -3\).

### Final Results
The solutions are:
1. For \(x = 4\), \(y = 7\), \(z = 3\).
2. For \(x = 7\), \(y = 4\), \(z = -3\).

### Summary of Relations
- In the first case: \(x < y\) and \(z > 0\).
- In the second case: \(x > y\) and \(z < 0\).