1. **State the problem:** We are given two linear equations: $$y = \frac{5}{9}x - 1$$ and $$y = -\frac{1}{9}x - 7$$ We want to find the point where these two lines intersect. 2. **Set the equations equal to find the intersection:** At the intersection point, both $y$ values are equal, so: $$\frac{5}{9}x - 1 = -\frac{1}{9}x - 7$$ 3. **Solve for $x$:** Add $\frac{1}{9}x$ to both sides: $$\frac{5}{9}x + \frac{1}{9}x - 1 = -7$$ Simplify the left side: $$\frac{6}{9}x - 1 = -7$$ Which simplifies to: $$\frac{2}{3}x - 1 = -7$$ Add 1 to both sides: $$\frac{2}{3}x = -7 + 1$$ $$\frac{2}{3}x = -6$$ Divide both sides by $\frac{2}{3}$: $$x = \frac{-6}{\cancel{\frac{2}{3}}} \times \cancel{\frac{3}{2}}$$ Since dividing by $\frac{2}{3}$ is the same as multiplying by $\frac{3}{2}$: $$x = -6 \times \frac{3}{2}$$ $$x = -9$$ 4. **Find $y$ by substituting $x = -9$ into one of the original equations:** Using the first equation: $$y = \frac{5}{9}(-9) - 1$$ $$y = -5 - 1$$ $$y = -6$$ 5. **Final answer:** The lines intersect at the point $$\boxed{(-9, -6)}$$.