1. **State the problem:** We are given that $m\angle A = 90^\circ$, $\cos B = 6x = 0.42$, and $\sin C = 10x = 0.34$. We need to solve for $x$. 2. **Analyze the given equations:** - From $\cos B = 6x = 0.42$, we interpret this as $6x = 0.42$. - From $\sin C = 10x = 0.34$, we interpret this as $10x = 0.34$. 3. **Solve each equation for $x$ separately:** - From $6x = 0.42$: $$x = \frac{0.42}{6}$$ $$x = 0.07$$ - From $10x = 0.34$: $$x = \frac{0.34}{10}$$ $$x = 0.034$$ 4. **Interpretation:** The two values for $x$ differ, so either the problem has inconsistent data or these represent different conditions. Since the problem asks to solve for $x$, we present both solutions. **Final answers:** $$x = 0.07 \quad \text{or} \quad x = 0.034$$