If x, y and z are three integers such that x + y = 8, y + z = 13 and z + x =17, then the value of ` (x^2)/(yz)` is :

Options

A. 1
B. `(18)/(11)`
C. `0`
D. `(7)/(5)`

Answer

To solve the problem step by step, we will use the equations provided and manipulate them to find the values of x, y, and z.

### Step 1: Write down the equations
We are given the following equations:
1. \( x + y = 8 \) (Equation 1)
2. \( y + z = 13 \) (Equation 2)
3. \( z + x = 17 \) (Equation 3)

### Step 2: Manipulate the equations
We can manipulate these equations to find the values of x, y, and z. Let's subtract Equation 1 from Equation 2.

From Equation 1:
\[ x + y = 8 \]

From Equation 2:
\[ y + z = 13 \]

Subtracting Equation 1 from Equation 2:
\[ (y + z) - (x + y) = 13 - 8 \]
This simplifies to:
\[ z - x = 5 \]
So, we can express this as:
\[ x - z = -5 \] (Equation 4)

### Step 3: Use Equation 4 with Equation 3
Now, we can add Equation 3 and Equation 4:
From Equation 3:
\[ z + x = 17 \]
From Equation 4:
\[ x - z = -5 \]

Adding these two equations:
\[ (z + x) + (x - z) = 17 - 5 \]
This simplifies to:
\[ 2x = 12 \]
So, we find:
\[ x = 6 \]

### Step 4: Substitute x back to find y and z
Now that we have \( x = 6 \), we can substitute this value back into Equation 1 to find y:
\[ x + y = 8 \]
\[ 6 + y = 8 \]
Thus:
\[ y = 8 - 6 = 2 \]

Next, we substitute \( y = 2 \) into Equation 2 to find z:
\[ y + z = 13 \]
\[ 2 + z = 13 \]
Thus:
\[ z = 13 - 2 = 11 \]

### Step 5: Calculate \( \frac{x^2}{yz} \)
Now we have the values:
- \( x = 6 \)
- \( y = 2 \)
- \( z = 11 \)

We need to calculate \( \frac{x^2}{yz} \):
\[ \frac{x^2}{yz} = \frac{6^2}{2 \cdot 11} \]
Calculating \( 6^2 \):
\[ 6^2 = 36 \]
Calculating \( 2 \cdot 11 \):
\[ 2 \cdot 11 = 22 \]
So:
\[ \frac{x^2}{yz} = \frac{36}{22} \]

### Step 6: Simplify the fraction
Now we simplify \( \frac{36}{22} \):
\[ \frac{36}{22} = \frac{18}{11} \]

### Final Answer
Thus, the value of \( \frac{x^2}{yz} \) is \( \frac{18}{11} \).

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