I. ` 8x+7y = 135" "` II.` 5x+6y = 99`
III.` 9y+8z= 121`

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 system of equations, we will follow these steps:

### Step 1: Write down the equations
We have the following equations:
1. \( 8x + 7y = 135 \) (Equation 1)
2. \( 5x + 6y = 99 \) (Equation 2)
3. \( 9y + 8z = 121 \) (Equation 3)

### Step 2: Solve Equations 1 and 2 for x and y
We can solve the first two equations simultaneously. Let's eliminate \( x \) from the equations.

First, we will multiply Equation 1 by 5 and Equation 2 by 8 to align the coefficients of \( x \):

- Multiply Equation 1 by 5:
\[ 5(8x + 7y) = 5(135) \implies 40x + 35y = 675 \quad \text{(Equation 4)} \]

- Multiply Equation 2 by 8:
\[ 8(5x + 6y) = 8(99) \implies 40x + 48y = 792 \quad \text{(Equation 5)} \]

### Step 3: Subtract Equation 4 from Equation 5
Now, we will subtract Equation 4 from Equation 5 to eliminate \( x \):
\[ (40x + 48y) - (40x + 35y) = 792 - 675 \]
This simplifies to:
\[ 48y - 35y = 117 \implies 13y = 117 \]

### Step 4: Solve for y
Now, divide both sides by 13:
\[ y = \frac{117}{13} = 9 \]

### Step 5: Substitute y back into one of the original equations to find x
We can substitute \( y = 9 \) back into Equation 1:
\[ 8x + 7(9) = 135 \]
This simplifies to:
\[ 8x + 63 = 135 \]
Subtract 63 from both sides:
\[ 8x = 135 - 63 \implies 8x = 72 \]
Now, divide by 8:
\[ x = \frac{72}{8} = 9 \]

### Step 6: Substitute x and y into Equation 3 to find z
Now that we have \( x = 9 \) and \( y = 9 \), we can substitute these values into Equation 3:
\[ 9(9) + 8z = 121 \]
This simplifies to:
\[ 81 + 8z = 121 \]
Subtract 81 from both sides:
\[ 8z = 121 - 81 \implies 8z = 40 \]
Now, divide by 8:
\[ z = \frac{40}{8} = 5 \]

### Final Values
Thus, we have:
- \( x = 9 \)
- \( y = 9 \)
- \( z = 5 \)

### Summary of Relationships
From the values obtained, we can see that:
- \( y = x \)
- \( y > z \)