1. **State the problem:** Solve the equation $$\sqrt{112} + y\sqrt{28} = 5\sqrt{7}$$ for $y$. 2. **Simplify the square roots:** - $$\sqrt{112} = \sqrt{16 \times 7} = 4\sqrt{7}$$ - $$\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}$$ 3. **Rewrite the equation using these simplifications:** $$4\sqrt{7} + y \times 2\sqrt{7} = 5\sqrt{7}$$ 4. **Factor out $\sqrt{7}$:** $$\sqrt{7}(4 + 2y) = 5\sqrt{7}$$ 5. **Divide both sides by $\sqrt{7}$:** $$\cancel{\sqrt{7}}(4 + 2y) = 5\cancel{\sqrt{7}}$$ $$4 + 2y = 5$$ 6. **Solve for $y$:** $$2y = 5 - 4$$ $$2y = 1$$ $$y = \frac{1}{2}$$ **Final answer:** $$y = \frac{1}{2}$$