If the operation $\wedge$ is defined by the equation $x \wedge y = 2x + y$, what is the value of $a$ in $2 \wedge a = a \wedge 3$?

### Concept & Formula This problem requires setting up a linear equation by expanding both sides of an equality using a provided custom operation rule. $$x \wedge y = 2x + y$$ ### Step-by-Step Solution * **Expand the left side ($2 \wedge a$):** Using the rule where $x = 2$ and $y = a$: $$2 \wedge a = 2(2) + a = 4 + a$$ * **Expand the right side ($a \wedge 3$):** Using the rule where $x = a$ and $y = 3$: $$a \wedge 3 = 2(a) + 3 = 2a + 3$$ * **Equate and solve for $a$:** The problem states that $2 \wedge a = a \wedge 3$. $$4 + a = 2a + 3$$ Subtract $a$ from both sides: $$4 = a + 3$$ Subtract 3 from both sides: $$1 = a$$ ### Exam Strategy & Shortcut For linear single-variable equations derived from custom operations, simply plug in the options directly into the expanded equation $4 + a = 2a + 3$ if the algebra feels slow. Test option (c) $a = 1$: $4 + 1 = 5$ on the left; $2(1) + 3 = 5$ on the right. Both sides match instantly. ### Common Pitfall Assuming the operation $\wedge$ is commutative (i.e., that $x \wedge y = y \wedge x$) and therefore concluding $a$ must be 2. The definition $2x + y$ heavily weights the first operand, meaning order matters immensely. ### Final Answer **Therefore, the correct answer is 1.**