If ` x-y +z = 5 and x^(2) +y^(2) +z^(2) = 49, ` find the value of ` : zx-xy -yz.`

Answer

To solve the problem, we start with the given equations:

1. \( x - y + z = 5 \)
2. \( x^2 + y^2 + z^2 = 49 \)

We need to find the value of \( zx - xy - yz \).

### Step 1: Use the identity for the square of a sum
We can use the identity:
\[ (x - y + z)^2 = x^2 + y^2 + z^2 + 2(zx - xy - yz) \]

### Step 2: Substitute the known values
From the first equation, we know:
\[ x - y + z = 5 \]
So, squaring both sides gives:
\[ (5)^2 = 25 \]

From the second equation, we know:
\[ x^2 + y^2 + z^2 = 49 \]

### Step 3: Substitute into the identity
Now substituting these values into the identity:
\[ 25 = 49 + 2(zx - xy - yz) \]

### Step 4: Rearrange the equation
Now, we rearrange the equation to isolate \( zx - xy - yz \):
\[ 25 - 49 = 2(zx - xy - yz) \]
\[ -24 = 2(zx - xy - yz) \]

### Step 5: Divide by 2
Now, divide both sides by 2 to solve for \( zx - xy - yz \):
\[ zx - xy - yz = \frac{-24}{2} = -12 \]

### Final Answer
Thus, the value of \( zx - xy - yz \) is:
\[ \boxed{-12} \]