1. **State the problem:** We need to find the solution to the system of equations: $$y = -x + 6$$ $$y = \frac{2}{3}x - 9$$ This means finding the point(s) where the two lines intersect. 2. **Set the equations equal to each other:** Since both equal $y$, set the right sides equal: $$-x + 6 = \frac{2}{3}x - 9$$ 3. **Solve for $x$:** Add $x$ to both sides: $$-x + x + 6 = \frac{2}{3}x + x - 9$$ $$6 = \frac{2}{3}x + x - 9$$ Rewrite $x$ as $\frac{3}{3}x$ to combine: $$6 = \frac{2}{3}x + \frac{3}{3}x - 9$$ $$6 = \frac{5}{3}x - 9$$ Add 9 to both sides: $$6 + 9 = \frac{5}{3}x - 9 + 9$$ $$15 = \frac{5}{3}x$$ Multiply both sides by the reciprocal $\frac{3}{5}$: $$x = 15 \times \frac{3}{5}$$ $$x = \cancel{15}^3 \times \frac{\cancel{3}}{5} = 9$$ 4. **Find $y$ by substituting $x=9$ into one of the original equations:** Using $y = -x + 6$: $$y = -9 + 6 = -3$$ 5. **Solution:** The lines intersect at the point $(9, -3)$. 6. **Check the solution:** Substitute $x=9$, $y=-3$ into the second equation: $$y = \frac{2}{3}x - 9$$ $$-3 = \frac{2}{3} \times 9 - 9$$ $$-3 = 6 - 9$$ $$-3 = -3$$ (True) **Final answer:** The solution to the system is **$(9, -3)$**.