1. **State the problem:** Solve the system of equations: $$x + 3y = -5$$ $$2x^2 + y^2 = 41$$ 2. **Express $x$ from the first equation:** $$x = -5 - 3y$$ 3. **Substitute $x$ into the second equation:** $$2(-5 - 3y)^2 + y^2 = 41$$ 4. **Expand the squared term:** $$(-5 - 3y)^2 = ( -3y - 5)^2 = 9y^2 + 30y + 25$$ 5. **Plug into the equation:** $$2(9y^2 + 30y + 25) + y^2 = 41$$ 6. **Distribute and simplify:** $$18y^2 + 60y + 50 + y^2 = 41$$ $$19y^2 + 60y + 50 = 41$$ 7. **Bring all terms to one side:** $$19y^2 + 60y + 9 = 0$$ 8. **Solve the quadratic equation for $y$ using the quadratic formula:** $$y = \frac{-60 \pm \sqrt{60^2 - 4 \cdot 19 \cdot 9}}{2 \cdot 19} = \frac{-60 \pm \sqrt{3600 - 684}}{38} = \frac{-60 \pm \sqrt{2916}}{38}$$ 9. **Simplify the square root:** $$\sqrt{2916} = 54$$ 10. **Calculate the two $y$ values:** $$y_1 = \frac{-60 + 54}{38} = \frac{-6}{38} = -\frac{3}{19}$$ $$y_2 = \frac{-60 - 54}{38} = \frac{-114}{38} = -3$$ 11. **Find corresponding $x$ values using $x = -5 - 3y$:** $$x_1 = -5 - 3\left(-\frac{3}{19}\right) = -5 + \frac{9}{19} = -\frac{95}{19} + \frac{9}{19} = -\frac{86}{19}$$ $$x_2 = -5 - 3(-3) = -5 + 9 = 4$$ **Final solutions:** $$(x, y) = \left(-\frac{86}{19}, -\frac{3}{19}\right) \text{ or } (4, -3)$$