If $x = y = 2z$ and $xyz = 256$, then $x$ is equal to

### Concept & Strategy This is an **Algebraic Substitution** problem. The strategy is to express all variables ($y$ and $z$) in terms of the target variable ($x$) using the given linear equality. This converts the multi-variable product equation into a single-variable cubic equation. ### Step-by-Step Solution * **Step 1: Express variables in terms of $x$** We are given $x = y = 2z$. From this, we can isolate $y$ and $z$ in terms of $x$: $$y = x$$ $$2z = x \Rightarrow z = \frac{x}{2}$$ * **Step 2: Substitute into the product equation** We are given $xyz = 256$. Substitute $y$ and $z$ with their $x$-equivalents: $$x \cdot (x) \cdot \left(\frac{x}{2}\right) = 256$$ * **Step 3: Solve for $x$** Multiply the terms: $$\frac{x^3}{2} = 256$$ Multiply both sides by 2: $$x^3 = 512$$ Take the cube root of both sides. Since $8 \times 8 \times 8 = 512$: $$x = 8$$ ### Exam Strategy & Shortcut For equations with neat integer options, **Option Verification** is incredibly fast. Test the options directly: If $x = 8$ (Option c), then $y = 8$. Since $2z = 8$, $z = 4$. Check the product: $8 \times 8 \times 4 = 64 \times 4 = 256$. It matches perfectly! You can bypass the algebraic solving entirely. ### Common Pitfall A very common mistake is solving for the wrong variable. Students might manipulate the equations to solve for $z$, find $z = 4$, see '4' in the options (Option b), and instantly select it without realizing the question explicitly asked for $x$. Always double-check what the question is requesting. ### Final Answer Therefore, the correct answer is **8**.