1. **State the problem:** Find the intersection point of the first pair of lines: $$y = -4x + 1$$ and $$y = 4x + 2$$. 2. **Set the equations equal to find the intersection:** Since both equal $$y$$, set $$-4x + 1 = 4x + 2$$. 3. **Solve for $$x$$:** $$-4x + 1 = 4x + 2$$ $$1 - 2 = 4x + 4x$$ $$-1 = 8x$$ $$x = -\frac{1}{8}$$ 4. **Find $$y$$ by substituting $$x$$ back into one of the equations:** Using $$y = -4x + 1$$: $$y = -4\left(-\frac{1}{8}\right) + 1 = \frac{1}{2} + 1 = \frac{3}{2}$$ 5. **Intersection point:** $$\left(-\frac{1}{8}, \frac{3}{2}\right)$$ --- 6. **State the problem:** Find the intersection point of the second pair: $$y = \frac{2}{3}x - 5$$ and $$y = \frac{2}{3}x + 6$$. 7. **Set equal:** $$\frac{2}{3}x - 5 = \frac{2}{3}x + 6$$ 8. **Simplify:** Subtract $$\frac{2}{3}x$$ from both sides: $$-5 = 6$$ This is a contradiction, so **no intersection**. These lines are parallel because they have the same slope but different intercepts. --- 9. **State the problem:** Find the intersection point of the third pair: $$y = \frac{2}{5}x - 3$$ and $$y = \frac{5}{2}x - 3$$. 10. **Set equal:** $$\frac{2}{5}x - 3 = \frac{5}{2}x - 3$$ 11. **Simplify:** Add 3 to both sides: $$\frac{2}{5}x = \frac{5}{2}x$$ 12. **Subtract $$\frac{2}{5}x$$ from both sides:** $$0 = \frac{5}{2}x - \frac{2}{5}x = \left(\frac{25}{10} - \frac{4}{10}\right)x = \frac{21}{10}x$$ 13. **Solve:** $$\frac{21}{10}x = 0 \implies x = 0$$ 14. **Find $$y$$:** Substitute $$x=0$$ into either equation: $$y = \frac{2}{5} \times 0 - 3 = -3$$ 15. **Intersection point:** $$\left(0, -3\right)$$ --- 16. **State the problem:** Find the intersection point of the fourth pair: $$y = -6x + 4$$ and $$y = \frac{1}{6}x - 7$$. 17. **Set equal:** $$-6x + 4 = \frac{1}{6}x - 7$$ 18. **Multiply both sides by 6 to clear denominator:** $$6(-6x + 4) = 6\left(\frac{1}{6}x - 7\right)$$ $$-36x + 24 = x - 42$$ 19. **Bring all terms to one side:** $$-36x - x = -42 - 24$$ $$-37x = -66$$ 20. **Solve for $$x$$:** $$x = \frac{66}{37}$$ 21. **Find $$y$$:** Substitute into $$y = -6x + 4$$: $$y = -6 \times \frac{66}{37} + 4 = -\frac{396}{37} + \frac{148}{37} = -\frac{248}{37}$$ 22. **Intersection point:** $$\left(\frac{66}{37}, -\frac{248}{37}\right)$$