1. **Problem Statement:** Solve the simultaneous equations: $$2x^2 + xy + y^2 = 14$$ $$x - y = 2$$ 2. **Step 1: Express one variable in terms of the other using the linear equation.** From $$x - y = 2$$, we get: $$x = y + 2$$ 3. **Step 2: Substitute $$x = y + 2$$ into the quadratic equation.** Substitute into $$2x^2 + xy + y^2 = 14$$: $$2(y+2)^2 + (y+2)y + y^2 = 14$$ 4. **Step 3: Expand and simplify.** $$2(y^2 + 4y + 4) + y^2 + 2y + y^2 = 14$$ $$2y^2 + 8y + 8 + y^2 + 2y + y^2 = 14$$ Combine like terms: $$2y^2 + y^2 + y^2 + 8y + 2y + 8 = 14$$ $$4y^2 + 10y + 8 = 14$$ 5. **Step 4: Bring all terms to one side to form a quadratic equation.** $$4y^2 + 10y + 8 - 14 = 0$$ $$4y^2 + 10y - 6 = 0$$ 6. **Step 5: Simplify the quadratic equation by dividing all terms by 2.** $$2y^2 + 5y - 3 = 0$$ 7. **Step 6: Solve the quadratic equation using the quadratic formula:** $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=2$$, $$b=5$$, $$c=-3$$. 8. **Step 7: Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4(2)(-3) = 25 + 24 = 49$$ 9. **Step 8: Find the roots:** $$y = \frac{-5 \pm \sqrt{49}}{2 \times 2} = \frac{-5 \pm 7}{4}$$ 10. **Step 9: Calculate each root:** - $$y_1 = \frac{-5 + 7}{4} = \frac{2}{4} = 0.5$$ - $$y_2 = \frac{-5 - 7}{4} = \frac{-12}{4} = -3$$ 11. **Step 10: Find corresponding $$x$$ values using $$x = y + 2$$:** - For $$y = 0.5$$, $$x = 0.5 + 2 = 2.5$$ - For $$y = -3$$, $$x = -3 + 2 = -1$$ 12. **Final answer:** The solutions to the simultaneous equations are: $$\boxed{(x, y) = (2.5, 0.5) \text{ or } (-1, -3)}$$