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 = sqrt((36)^((1)/(2)) xx (1296)^((1)/(4)))`
II. ` 2 y + 3 z = 33 `
III. ` 6 y + 5 z = 71 `

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 lt z `

Answer

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

### Step 1: Solve for \( x \) from Equation (I)
Given:
\[ x = \sqrt{(36)^{\frac{1}{2}} \times (1296)^{\frac{1}{4}}} \]

First, simplify \( (36)^{\frac{1}{2}} \):
\[ (36)^{\frac{1}{2}} = 6 \]

Next, simplify \( (1296)^{\frac{1}{4}} \):
\[ 1296 = 6^4 \quad \text{(since } 6 \times 6 \times 6 \times 6 = 1296\text{)} \]
Thus,
\[ (1296)^{\frac{1}{4}} = 6 \]

Now substituting back:
\[ x = \sqrt{6 \times 6} = \sqrt{36} = 6 \]

### Step 2: Use Equations (II) and (III) to find \( y \) and \( z \)
We have:
1. \( 2y + 3z = 33 \) (Equation II)
2. \( 6y + 5z = 71 \) (Equation III)

### Step 3: Multiply Equation (II) to align coefficients
Multiply Equation (II) by 3:
\[ 3(2y + 3z) = 3(33) \]
This gives us:
\[ 6y + 9z = 99 \quad \text{(Equation IV)} \]

### Step 4: Subtract Equation (III) from Equation (IV)
Now subtract Equation (III) from Equation (IV):
\[ (6y + 9z) - (6y + 5z) = 99 - 71 \]
This simplifies to:
\[ 4z = 28 \]
Thus,
\[ z = \frac{28}{4} = 7 \]

### Step 5: Substitute \( z \) back into Equation (II) to find \( y \)
Substituting \( z = 7 \) into Equation (II):
\[ 2y + 3(7) = 33 \]
This simplifies to:
\[ 2y + 21 = 33 \]
Subtract 21 from both sides:
\[ 2y = 12 \]
Thus,
\[ y = \frac{12}{2} = 6 \]

### Step 6: Summarize the values
Now we have:
- \( x = 6 \)
- \( y = 6 \)
- \( z = 7 \)

### Step 7: Determine the relationship
Now we can compare the values:
- \( x = y = 6 \)
- \( z = 7 \)

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

### Final Answer
The relationship between \( x \), \( y \), and \( z \) is \( x = y < z \).

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