1. **State the problem:** Solve the system of equations: $$y = 2x + 6$$ $$16x^2 + y^2 = 36$$ 2. **Substitute** the expression for $y$ from the first equation into the second equation: $$16x^2 + (2x + 6)^2 = 36$$ 3. **Expand** the squared term: $$(2x + 6)^2 = 4x^2 + 24x + 36$$ So the equation becomes: $$16x^2 + 4x^2 + 24x + 36 = 36$$ 4. **Combine like terms:** $$20x^2 + 24x + 36 = 36$$ 5. **Subtract 36 from both sides:** $$20x^2 + 24x + 36 - 36 = 36 - 36$$ $$20x^2 + 24x = 0$$ 6. **Factor out the common factor:** $$4x(5x + 6) = 0$$ 7. **Set each factor equal to zero:** $$4x = 0 \implies x = 0$$ $$5x + 6 = 0 \implies 5x = -6 \implies x = -\frac{6}{5}$$ 8. **Find corresponding $y$ values using $y = 2x + 6$:** For $x=0$: $$y = 2(0) + 6 = 6$$ For $x = -\frac{6}{5}$: $$y = 2\left(-\frac{6}{5}\right) + 6 = -\frac{12}{5} + 6 = -\frac{12}{5} + \frac{30}{5} = \frac{18}{5}$$ 9. **List all solutions:** $$\boxed{(0,6) \text{ and } \left(-\frac{6}{5}, \frac{18}{5}\right)}$$