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

Options

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

Answer

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

### Step 1: Solve the first two equations for \(x\) and \(y\)

We have the following equations:

1. \(8x + 7y = 135\) (Equation I)
2. \(5x + 6y = 99\) (Equation II)

We can add these two equations together.

\[ (8x + 7y) + (5x + 6y) = 135 + 99 \]

This simplifies to:

\[ (8 + 5)x + (7 + 6)y = 234 \]

\[ 13x + 13y = 234 \]

Now, we can factor out the common term:

\[ 13(x + y) = 234 \]

Dividing both sides by 13 gives us:

\[ x + y = 18 \quad \text{(Equation IV)} \]

### Step 2: Substitute \(x + y\) into one of the original equations

Now, we can use Equation IV to find \(y\) in terms of \(x\). We can substitute \(y = 18 - x\) into Equation II:

\[ 5x + 6(18 - x) = 99 \]

Expanding this gives:

\[ 5x + 108 - 6x = 99 \]

Combining like terms results in:

\[ -x + 108 = 99 \]

Subtracting 108 from both sides:

\[ -x = 99 - 108 \]

\[ -x = -9 \]

Thus, we find:

\[ x = 9 \]

### Step 3: Find \(y\)

Using the value of \(x\) in Equation IV:

\[ x + y = 18 \]

Substituting \(x = 9\):

\[ 9 + y = 18 \]

Solving for \(y\):

\[ y = 18 - 9 = 9 \]

### Step 4: Find \(z\) using the third equation

Now we will use the value of \(y\) to find \(z\) using Equation III:

\[ 9y + 8z = 121 \]

Substituting \(y = 9\):

\[ 9(9) + 8z = 121 \]

Calculating \(9(9)\):

\[ 81 + 8z = 121 \]

Subtracting 81 from both sides:

\[ 8z = 121 - 81 \]

\[ 8z = 40 \]

Dividing both sides by 8 gives:

\[ z = 5 \]

### Step 5: Summarize the results

We have found:

- \(x = 9\)
- \(y = 9\)
- \(z = 5\)

### Step 6: Determine the relationship between \(x\), \(y\), and \(z\)

Now we can compare the values:

- \(x = y = 9\)
- \(z = 5\)

Thus, the relationship is:

\[ x = y > z \]

### Final Answer

The final relation is \(x = y > z\).

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