1. **State the problem:** Solve the simultaneous equations: $$y = 4 - x$$ $$x^2 + 2y^2 = 67$$ 2. **Substitute** the expression for $y$ from the first equation into the second equation: $$x^2 + 2(4 - x)^2 = 67$$ 3. **Expand** the squared term: $$(4 - x)^2 = 16 - 8x + x^2$$ So the equation becomes: $$x^2 + 2(16 - 8x + x^2) = 67$$ 4. **Distribute** the 2: $$x^2 + 32 - 16x + 2x^2 = 67$$ 5. **Combine like terms:** $$3x^2 - 16x + 32 = 67$$ 6. **Bring all terms to one side:** $$3x^2 - 16x + 32 - 67 = 0$$ $$3x^2 - 16x - 35 = 0$$ 7. **Solve the quadratic equation** using the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=-16$, and $c=-35$. Calculate the discriminant: $$\Delta = (-16)^2 - 4 \times 3 \times (-35) = 256 + 420 = 676$$ 8. **Find the roots:** $$x = \frac{16 \pm \sqrt{676}}{6} = \frac{16 \pm 26}{6}$$ So, $$x_1 = \frac{16 + 26}{6} = \frac{42}{6} = 7$$ $$x_2 = \frac{16 - 26}{6} = \frac{-10}{6} = -\frac{5}{3}$$ 9. **Find corresponding $y$ values** using $y = 4 - x$: For $x=7$: $$y = 4 - 7 = -3$$ For $x = -\frac{5}{3}$: $$y = 4 - \left(-\frac{5}{3}\right) = 4 + \frac{5}{3} = \frac{12}{3} + \frac{5}{3} = \frac{17}{3}$$ **Final solutions:** $$(x, y) = (7, -3) \quad \text{or} \quad \left(-\frac{5}{3}, \frac{17}{3}\right)$$