If $13 = \frac{13w}{(1 - w)}$, then $(2w)^2 = x$

### Concept & Formula This question requires solving a linear equation for a variable and then substituting that value into a secondary algebraic expression. ### Step-by-Step Solution 1. Start with the initial equation: $13 = \frac{13w}{1 - w}$. 2. Divide both sides by 13 to simplify the equation: $1 = \frac{w}{1 - w}$. 3. Multiply both sides by the denominator $(1 - w)$ to eliminate the fraction: $1(1 - w) = w$. 4. This simplifies to: $1 - w = w$. 5. Add $w$ to both sides to group the variables: $1 = 2w$. 6. The question asks for the value of $(2w)^2$. 7. Instead of solving for $w$, substitute the entire expression $2w = 1$ directly into the target expression: $(1)^2$. 8. Calculate the final value: $1^2 = 1$. ### Exam Strategy & Shortcut Don't blindly solve for $w$ down to $w = 0.5$. Always keep an eye on the target expression, which in this case is $(2w)^2$. When you reach the step $2w = 1$, you can stop algebraic manipulation immediately and square the result to get 1. ### Common Pitfall Students often solve all the way to $w = 1/2$, plug it back into the formula $(2 \times 1/2)^2$, and make a careless arithmetic mistake under time pressure (like writing $1/4$ by improperly distributing the exponent). ### Final Answer **Therefore, the correct answer is 1.**