Step 1: Substitute the Values
To solve the expression \( \frac{X + Y + Z}{XY + YZ + ZX} \), we first substitute the given values:
- \( X = 2 \)
- \( Y = -1 \)
- \( Z = 3 \)

Step 2: Calculate the Numerator
Now, calculate the numerator \( X + Y + Z \):
- \( 2 + (-1) + 3 \)
- \( = 2 - 1 + 3 \)
- \( = 4 \)

Step 3: Calculate the Denominator
Next, calculate the denominator \( XY + YZ + ZX \):
- \( XY = 2 \times (-1) = -2 \)
- \( YZ = (-1) \times 3 = -3 \)
- \( ZX = 3 \times 2 = 6 \)
Now, sum these values:
- \( -2 + (-3) + 6 \)
- \( = -2 - 3 + 6 \)
- \( = 1 \)

Step 4: Put it All Together
Now we can substitute the results back into the fraction:
- \( \frac{4}{1} = 4 \)

Final Answer
Thus, the value of the expression \( \frac{X + Y + Z}{XY + YZ + ZX} \) when \( X = 2, Y = -1, Z = 3 \) is:
- **4**
This detailed breakdown shows the substitution, calculation of the numerator and denominator, and the final result.