1. **State the problem:** Solve the equation $x\sqrt{2} = 7\sqrt{2}$ for $x$. 2. **Formula and rules:** To solve for $x$, we want to isolate $x$ on one side of the equation. Since both sides are multiplied by $\sqrt{2}$, we can divide both sides by $\sqrt{2}$ to cancel it out. 3. **Intermediate work:** $$x\sqrt{2} = 7\sqrt{2}$$ Divide both sides by $\sqrt{2}$: $$\frac{x\sqrt{2}}{\cancel{\sqrt{2}}} = \frac{7\sqrt{2}}{\cancel{\sqrt{2}}}$$ 4. **Simplify:** $$x = 7$$ 5. **Explanation:** Dividing both sides by the same nonzero quantity $\sqrt{2}$ keeps the equation balanced. The $\sqrt{2}$ cancels out, leaving $x = 7$. **Final answer:** $$x = 7$$