1. **State the problem:** We need to solve the simultaneous equations graphically: $$y = x^2 - x - x - 6$$ and $$y = 3x - 2$$ 2. **Simplify the first equation:** Combine like terms in the quadratic equation: $$y = x^2 - 2x - 6$$ 3. **Understand the equations:** - The first equation is a quadratic function (a parabola). - The second equation is a linear function (a straight line). 4. **Graphical solution:** The solutions to the system are the points where the parabola and the line intersect. 5. **Set the equations equal to find intersection points algebraically:** $$x^2 - 2x - 6 = 3x - 2$$ 6. **Bring all terms to one side:** $$x^2 - 2x - 6 - 3x + 2 = 0$$ $$x^2 - 5x - 4 = 0$$ 7. **Solve the quadratic equation:** Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-5$, $c=-4$. Calculate the discriminant: $$\Delta = (-5)^2 - 4(1)(-4) = 25 + 16 = 41$$ Calculate the roots: $$x = \frac{5 \pm \sqrt{41}}{2}$$ 8. **Find corresponding y-values:** Use the linear equation $y = 3x - 2$: $$y_1 = 3 \times \frac{5 + \sqrt{41}}{2} - 2$$ $$y_2 = 3 \times \frac{5 - \sqrt{41}}{2} - 2$$ 9. **Final answer:** The solutions are the points: $$\left( \frac{5 + \sqrt{41}}{2}, 3 \times \frac{5 + \sqrt{41}}{2} - 2 \right)$$ and $$\left( \frac{5 - \sqrt{41}}{2}, 3 \times \frac{5 - \sqrt{41}}{2} - 2 \right)$$