If ` 2 x + 3y - 5 z = 18 , 3 x + 2y + z = 29 and x + y + 3z = 17` then what is the value of ` xy + yz + zx` ?

Options

A. 32
B. 52
C. 64
D. 46

Answer

To solve the system of equations and find the value of \( xy + yz + zx \), we will follow these steps:

Given equations:
1. \( 2x + 3y - 5z = 18 \) (Equation 1)
2. \( 3x + 2y + z = 29 \) (Equation 2)
3. \( x + y + 3z = 17 \) (Equation 3)

### Step 1: Eliminate one variable
We will eliminate \( x \) by manipulating Equations 2 and 3. First, we multiply Equation 3 by 3:

\[ 3(x + y + 3z) = 3 \cdot 17 \]
This gives us:
\[ 3x + 3y + 9z = 51 \quad \text{(Equation 4)} \]

### Step 2: Subtract Equation 2 from Equation 4
Now, we subtract Equation 2 from Equation 4:

\[ (3x + 3y + 9z) - (3x + 2y + z) = 51 - 29 \]
This simplifies to:
\[ y + 8z = 22 \quad \text{(Equation 5)} \]

### Step 3: Eliminate \( x \) from Equations 1 and 3
Next, we will eliminate \( x \) from Equations 1 and 3. We multiply Equation 3 by 2:

\[ 2(x + y + 3z) = 2 \cdot 17 \]
This gives us:
\[ 2x + 2y + 6z = 34 \quad \text{(Equation 6)} \]

Now, we subtract Equation 1 from Equation 6:

\[ (2x + 2y + 6z) - (2x + 3y - 5z) = 34 - 18 \]
This simplifies to:
\[ -y + 11z = 16 \quad \text{(Equation 7)} \]

### Step 4: Solve Equations 5 and 7
Now we have two equations (5 and 7) with two variables \( y \) and \( z \):
1. \( y + 8z = 22 \) (Equation 5)
2. \( -y + 11z = 16 \) (Equation 7)

Adding these two equations:

\[ (y + 8z) + (-y + 11z) = 22 + 16 \]
This simplifies to:
\[ 19z = 38 \]
Thus, we find:
\[ z = \frac{38}{19} = 2 \]

### Step 5: Substitute \( z \) back to find \( y \)
Now, substitute \( z = 2 \) back into Equation 5:

\[ y + 8(2) = 22 \]
This simplifies to:
\[ y + 16 = 22 \]
Thus, we find:
\[ y = 22 - 16 = 6 \]

### Step 6: Substitute \( y \) and \( z \) back to find \( x \)
Now, substitute \( y = 6 \) and \( z = 2 \) back into Equation 3:

\[ x + 6 + 3(2) = 17 \]
This simplifies to:
\[ x + 6 + 6 = 17 \]
Thus, we find:
\[ x + 12 = 17 \implies x = 17 - 12 = 5 \]

### Step 7: Calculate \( xy + yz + zx \)
Now we have \( x = 5 \), \( y = 6 \), and \( z = 2 \). We can now calculate \( xy + yz + zx \):

\[ xy = 5 \cdot 6 = 30 \]
\[ yz = 6 \cdot 2 = 12 \]
\[ zx = 2 \cdot 5 = 10 \]

Adding these together:

\[ xy + yz + zx = 30 + 12 + 10 = 52 \]

### Final Answer
The value of \( xy + yz + zx \) is \( \boxed{52} \).

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