1. **State the problem:** We are given two equations: $$y = \frac{1}{3}x + 3$$ and $$5x + 3y = -9$$ We want to analyze these equations, find their relationship, and possibly find their intersection. 2. **Rewrite the second equation in slope-intercept form:** Start with: $$5x + 3y = -9$$ Subtract $5x$ from both sides: $$\cancel{5x} + 3y - \cancel{5x} = -9 - 5x$$ which simplifies to: $$3y = -9 - 5x$$ Divide both sides by 3: $$y = \frac{-9 - 5x}{3} = -3 - \frac{5}{3}x$$ 3. **Compare the two lines:** First line: $$y = \frac{1}{3}x + 3$$ Second line: $$y = -\frac{5}{3}x - 3$$ 4. **Interpretation:** The slopes are different ($\frac{1}{3}$ vs $-\frac{5}{3}$), so the lines are not parallel and will intersect at exactly one point. 5. **Find the intersection point:** Set the two expressions for $y$ equal: $$\frac{1}{3}x + 3 = -\frac{5}{3}x - 3$$ Add $\frac{5}{3}x$ to both sides: $$\frac{1}{3}x + \frac{5}{3}x + 3 = -3$$ Combine like terms: $$\frac{6}{3}x + 3 = -3$$ Simplify: $$2x + 3 = -3$$ Subtract 3 from both sides: $$2x = -6$$ Divide both sides by 2: $$x = -3$$ 6. **Find $y$ coordinate:** Substitute $x = -3$ into the first equation: $$y = \frac{1}{3}(-3) + 3 = -1 + 3 = 2$$ 7. **Final answer:** The lines intersect at the point $$(-3, 2)$$.