1. **State the problem:** Solve the simultaneous equations: $$x - 6y = 10$$ $$3y^2 = 4x + 7$$ 2. **Show that** $3y^2 - 24y - 47 = 0$: From the first equation, express $x$ in terms of $y$: $$x = 6y + 10$$ Substitute into the second equation: $$3y^2 = 4(6y + 10) + 7$$ $$3y^2 = 24y + 40 + 7$$ $$3y^2 = 24y + 47$$ Bring all terms to one side: $$3y^2 - 24y - 47 = 0$$ 3. **Solve the quadratic equation** $3y^2 - 24y - 47 = 0$: Use the quadratic formula: $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=3$, $b=-24$, $c=-47$. Calculate the discriminant: $$\Delta = (-24)^2 - 4 \times 3 \times (-47) = 576 + 564 = 1140$$ Calculate the roots: $$y = \frac{24 \pm \sqrt{1140}}{6}$$ Approximate $\sqrt{1140} \approx 33.76$: $$y_1 = \frac{24 + 33.76}{6} = \frac{57.76}{6} = 9.6$$ $$y_2 = \frac{24 - 33.76}{6} = \frac{-9.76}{6} = -1.6$$ 4. **Find corresponding $x$ values** using $x = 6y + 10$: For $y=9.6$: $$x = 6 \times 9.6 + 10 = 57.6 + 10 = 67.6$$ For $y=-1.6$: $$x = 6 \times (-1.6) + 10 = -9.6 + 10 = 0.4$$ 5. **Final answers to 1 decimal place:** $$(x,y) = (67.6, 9.6) \text{ or } (0.4, -1.6)$$