If $x + y + z = 9$ and both $y$ and $z$ are positive integers greater than zero, then the maximum value $x$ can take is

### Concept & Logic In an equation with multiple variables summing to a constant, to maximize one variable, you must minimize all other variables subject to the given boundary constraints. ### Step-by-Step Solution **Given:** * $x + y + z = 9$ * $y$ and $z$ are strictly positive integers ($y > 0$, $z > 0$). **Calculation / Deduction:** To find the maximum possible value for $x$, we must assign the absolute minimum possible values to $y$ and $z$. Since $y$ and $z$ are positive integers, the smallest integer value they can take is $1$. Let $y = 1$ and $z = 1$. Substitute these minimum values into the equation: $$x + 1 + 1 = 9$$ $$x + 2 = 9$$ $$x = 7$$ ### Exam Strategy & Shortcut When dealing with "maximize one term in a sum" problems, immediately peg the other terms to their absolute minimum boundary conditions. Positive integer means minimum is $1$. So, $9 - 1 - 1 = 7$. This can be solved mentally in under 5 seconds. ### Common Pitfall The most frequent mistake is assuming "positive integer" includes $0$, leading to $y=0$, $z=0$, and $x=9$, which is incorrect. Zero is a neutral integer—neither positive nor negative. Always remember positive integers strictly start from $1$. ### Final Answer Therefore, the correct answer is 7.