If $2x + y = 5$ and $3x - 4y = 2$, then the value of $2xy$ is

### Concept & Formula To solve a system of two linear equations, we use substitution or elimination to find the individual values of $x$ and $y$. Once identified, we simply substitute them into the target expression, $2xy$. $$ ax + by = c $$ ### Step-by-Step Solution **Given:** 1. $2x + y = 5$ 2. $3x - 4y = 2$ **Calculation:** * From equation (1), isolate $y$: $y = 5 - 2x$ * Substitute this expression for $y$ into equation (2): $3x - 4(5 - 2x) = 2$ * Distribute the $-4$: $3x - 20 + 8x = 2$ * Combine like terms and solve for $x$: $11x = 22$ $x = 2$ * Substitute $x = 2$ back into our isolated $y$ equation: $y = 5 - 2(2)$ $y = 5 - 4 = 1$ * Finally, compute the requested value of $2xy$: $2xy = 2(2)(1) = 4$ ### Exam Strategy & Shortcut Instead of substitution, the Elimination Method is often faster for avoiding sign errors. Multiply the first equation by 4 to quickly eliminate $y$: $8x + 4y = 20$ $3x - 4y = 2$ Add them together vertically: $11x = 22 \Rightarrow x = 2$. This requires fewer writing steps on rough paper during high-pressure exams. ### Common Pitfall A major trap is solving for $x = 2$ or $y = 1$ and frantically looking for those numbers in the options, or forgetting to multiply them by the coefficient 2 in the final step ($2xy$). Always re-read the exact variable expression the question is asking for. ### Final Answer **Therefore, the correct answer is 4.**