1. **State the problem:** We have two zoo membership plans with costs depending on the number of family members $x$. Plan A costs $y = 20x + 15$ and Plan B costs $y = 15x + 30$. We want to understand these cost functions and find where they intersect. 2. **Formulas and rules:** Both plans are linear functions of the form $y = mx + b$, where $m$ is the slope (cost per family member) and $b$ is the fixed fee. 3. **Set the two plans equal to find the intersection point:** $$20x + 15 = 15x + 30$$ 4. **Solve for $x$:** $$20x + 15 = 15x + 30$$ $$20x - 15x = 30 - 15$$ $$5x = 15$$ $$x = \frac{\cancel{5}x}{\cancel{5}} = \frac{15}{5} = 3$$ 5. **Find the cost at $x=3$ for either plan:** For Plan A: $$y = 20(3) + 15 = 60 + 15 = 75$$ 6. **Interpretation:** At 3 family members, both plans cost the same, $75$. For fewer than 3 members, Plan B is cheaper; for more than 3 members, Plan A is cheaper. 7. **Summary:** - Plan A: $y = 20x + 15$ (steeper slope, lower fixed fee) - Plan B: $y = 15x + 30$ (less steep slope, higher fixed fee) - Intersection at $(3, 75)$ where costs are equal. This helps decide which plan is more cost-effective depending on family size.