If $3x + 7y = 75$ and $5x - 5y = 25$, then what is the value of $x + y$?

### Concept & Strategy This is a standard **System of Linear Equations** problem. The most effective strategy is to look for common factors in the equations to simplify them before attempting elimination or substitution. ### Step-by-Step Solution * **Step 1: Simplify the second equation** Look closely at $5x - 5y = 25$. All coefficients are multiples of 5. Divide the entire equation by 5: $$x - y = 5$$ From this, we can easily isolate $x$: $$x = y + 5$$ * **Step 2: Substitute into the first equation** The first equation is $3x + 7y = 75$. Substitute $x = (y + 5)$ into this equation: $$3(y + 5) + 7y = 75$$ $$3y + 15 + 7y = 75$$ Combine the $y$ terms: $$10y + 15 = 75$$ $$10y = 60$$ $$y = 6$$ * **Step 3: Solve for $x$** Substitute $y = 6$ back into our simplified equation $x = y + 5$: $$x = 6 + 5$$ $$x = 11$$ * **Step 4: Calculate the final required value** The question asks for $x + y$: $$x + y = 11 + 6 = 17$$ ### Exam Strategy & Shortcut Always look for an equation that can be reduced! If you immediately jump to cross-multiplying (e.g., multiplying the first equation by 5 and the second by 3), you will end up dealing with large, error-prone numbers ($15x + 35y = 375$). Spotting that $5x - 5y = 25$ reduces to $x - y = 5$ cuts the calculation time in half. ### Common Pitfall A common error is stopping the calculation prematurely. After working through the algebra to find $x = 11$ or $y = 6$, students sometimes forget that the question specifically asks for their sum ($x + y$). Always re-read the final line of the question before marking your choice. ### Final Answer Therefore, the correct answer is **17**.