I. ` 7x + 6y = 110`
II. ` 4x + 3y = 59`
III.` X + z = 15`

Options

A. `x lt y = z `
B. ` x le y lt z `
C. ` x lt y gt z `
D. ` x = y gt z `

Explanation

To solve the given equations step by step, we will follow the process of substitution and elimination.

### Step 1: Write down the equations
We have the following equations:
1. \( 7x + 6y = 110 \) (Equation 1)
2. \( 4x + 3y = 59 \) (Equation 2)
3. \( x + z = 15 \) (Equation 3)

### Step 2: Solve the first two equations
To eliminate \( y \), we can manipulate the second equation. Let's multiply Equation 2 by 2:
\[ 2(4x + 3y) = 2(59) \implies 8x + 6y = 118 \quad (Equation 4) \]

### Step 3: Subtract Equation 1 from Equation 4
Now, we will subtract Equation 1 from Equation 4:
\[ (8x + 6y) - (7x + 6y) = 118 - 110 \]
This simplifies to:
\[ 8x - 7x + 6y - 6y = 8 \implies x = 8 \]

### Step 4: Substitute \( x \) back to find \( y \)
Now that we have \( x = 8 \), we can substitute this value back into Equation 2 to find \( y \):
\[ 4(8) + 3y = 59 \implies 32 + 3y = 59 \]
Subtract 32 from both sides:
\[ 3y = 59 - 32 \implies 3y = 27 \]
Now divide by 3:
\[ y = 9 \]

### Step 5: Substitute \( x \) to find \( z \)
Now, we can use the value of \( x \) to find \( z \) using Equation 3:
\[ x + z = 15 \implies 8 + z = 15 \]
Subtract 8 from both sides:
\[ z = 15 - 8 \implies z = 7 \]

### Step 6: Summary of values
We have found:
- \( x = 8 \)
- \( y = 9 \)
- \( z = 7 \)

### Step 7: Determine the relationship between \( x, y, z \)
Now, let's compare the values:
- \( z = 7 \)
- \( x = 8 \)
- \( y = 9 \)

From this, we can see:
- \( z < x < y \)

### Final Relation
Thus, the relationship can be expressed as:
\[ z < x < y \]