1. **State the problem:** Solve the equation $w - 3 = \sqrt{5w - 19}$ for $w$. 2. **Understand the equation:** The equation involves a square root. To solve it, we will isolate the square root and then square both sides to eliminate it. 3. **Isolate the square root:** The equation is already isolated as $w - 3 = \sqrt{5w - 19}$. 4. **Square both sides:** $$ (w - 3)^2 = (\sqrt{5w - 19})^2 $$ $$ (w - 3)^2 = 5w - 19 $$ 5. **Expand the left side:** $$ (w - 3)^2 = w^2 - 6w + 9 $$ 6. **Set up the quadratic equation:** $$ w^2 - 6w + 9 = 5w - 19 $$ 7. **Bring all terms to one side:** $$ w^2 - 6w + 9 - 5w + 19 = 0 $$ $$ w^2 - 11w + 28 = 0 $$ 8. **Factor the quadratic:** $$ (w - 7)(w - 4) = 0 $$ 9. **Solve for $w$:** $$ w = 7 \quad \text{or} \quad w = 4 $$ 10. **Check for extraneous solutions:** Substitute back into the original equation. - For $w=7$: $$ 7 - 3 = 4 $$ $$ \sqrt{5(7) - 19} = \sqrt{35 - 19} = \sqrt{16} = 4 $$ Valid. - For $w=4$: $$ 4 - 3 = 1 $$ $$ \sqrt{5(4) - 19} = \sqrt{20 - 19} = \sqrt{1} = 1 $$ Valid. **Final answer:** $$ w = 7 \quad \text{or} \quad w = 4 $$