Three equations (I), (II) and (III) are given in each question. On the basis of these equations you have to decide the relation between 'x' , 'y' and 'z' and give answer
I. ` x + 3y + 4 z = 96`
II. ` 2 x + 8 z = 80`
III. ` 2 x + 6y = 120`

Options

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

Answer

To solve the given equations and find the relationship between \( x \), \( y \), and \( z \), we will follow these steps:

### Step 1: Write down the equations
We have the following equations:
1. \( x + 3y + 4z = 96 \) (Equation I)
2. \( 2x + 8z = 80 \) (Equation II)
3. \( 2x + 6y = 120 \) (Equation III)

### Step 2: Manipulate Equation I
Multiply Equation I by 2 to facilitate comparison with the other equations:
\[ 2(x + 3y + 4z) = 2(96) \]
This simplifies to:
\[ 2x + 6y + 8z = 192 \quad \text{(Equation IV)} \]

### Step 3: Set up the equations for comparison
Now we have:
- Equation IV: \( 2x + 6y + 8z = 192 \)
- Equation II: \( 2x + 8z = 80 \)
- Equation III: \( 2x + 6y = 120 \)

### Step 4: Substitute Equation III into Equation IV
From Equation III, we know:
\[ 2x + 6y = 120 \]
Substituting this into Equation IV:
\[ 120 + 8z = 192 \]
Now, isolate \( z \):
\[ 8z = 192 - 120 \]
\[ 8z = 72 \]
\[ z = \frac{72}{8} = 9 \]

### Step 5: Substitute the value of \( z \) into Equation II
Now that we have \( z = 9 \), we can substitute this value back into Equation II to find \( x \):
\[ 2x + 8(9) = 80 \]
\[ 2x + 72 = 80 \]
\[ 2x = 80 - 72 \]
\[ 2x = 8 \]
\[ x = \frac{8}{2} = 4 \]

### Step 6: Substitute the value of \( x \) into Equation III
Now, we substitute \( x = 4 \) back into Equation III to find \( y \):
\[ 2(4) + 6y = 120 \]
\[ 8 + 6y = 120 \]
\[ 6y = 120 - 8 \]
\[ 6y = 112 \]
\[ y = \frac{112}{6} = \frac{56}{3} \approx 18.67 \]

### Step 7: Determine the relationship between \( x \), \( y \), and \( z \)
We have:
- \( x = 4 \)
- \( y = \frac{56}{3} \approx 18.67 \)
- \( z = 9 \)

Now we can compare the values:
- \( x < z \) (since \( 4 < 9 \))
- \( y > z \) (since \( \frac{56}{3} \approx 18.67 > 9 \))
- \( x < y \) (since \( 4 < \frac{56}{3} \))

Thus, the relationship is:
\[ x < y > z \]

### Final Answer
The relationship between \( x \), \( y \), and \( z \) is:
\[ x < y > z \]