If $(x + y)^2 - z^2 = 4$, $(y + z)^2 - x^2 = 9$, $(z + x)^2 - y^2 = 36$, what is/are the value(s) of $x + y + z$?

### Concept & Formula This problem features a system of equations linked by the **Difference of Squares** identity. Recognizing this pattern allows us to factor each equation and extract a common multiplier that simplifies the entire system. $$A^2 - B^2 = (A - B)(A + B)$$ ### Step-by-Step Solution Given the system of equations: 1) $(x + y)^2 - z^2 = 4$ 2) $(y + z)^2 - x^2 = 9$ 3) $(z + x)^2 - y^2 = 36$ **Step 1: Factor each equation** Apply the difference of squares identity to each line. Notice that the $(A+B)$ term for every single equation will become $(x+y+z)$. 1) $((x + y) - z)((x + y) + z) = 4 \implies (x + y - z)(x + y + z) = 4$ 2) $((y + z) - x)((y + z) + x) = 9 \implies (y + z - x)(x + y + z) = 9$ 3) $((z + x) - y)((z + x) + y) = 36 \implies (z + x - y)(x + y + z) = 36$ **Step 2: Add all equations together** To leverage the common $(x+y+z)$ term, add equations 1, 2, and 3 vertically: $(x + y - z)(x + y + z) + (y + z - x)(x + y + z) + (z + x - y)(x + y + z) = 4 + 9 + 36$ Factor out the common $(x + y + z)$: $(x + y + z) \cdot [ (x + y - z) + (y + z - x) + (z + x - y) ] = 49$ **Step 3: Simplify the bracketed terms** Combine the terms inside the square brackets: $x + y - z + y + z - x + z + x - y$ - $x - x + x = x$ - $y + y - y = y$ - $-z + z + z = z$ The bracket simplifies beautifully to $(x + y + z)$. **Step 4: Solve for $(x+y+z)$** Substitute the simplified bracket back into the equation: $(x + y + z) \cdot (x + y + z) = 49$ $(x + y + z)^2 = 49$ Take the square root of both sides. Remember that squaring a negative number yields a positive result. $x + y + z = \pm 7$ ### Exam Strategy & Shortcut **Pattern Recognition (The Addition Trick):** When you see cyclical differences of squares equal to constants, the sum of those constants is almost always the square of the sum of the variables. $4 + 9 + 36 = 49$. The square root of 49 is $\pm 7$. You can literally solve this 3-variable quadratic system in 10 seconds in your head. ### Common Pitfall Forgetting the $\pm$ when taking the square root. Students often see $\sqrt{49}$ and immediately look for positive 7, missing that $(-7)^2$ is also 49. ### Final Answer Therefore, the correct answer is **$\pm 7$**.