1. **State the problem:** We need to find the number of people $n$ who can participate in each activity for exactly 29 dollars. 2. **Write the equations:** - Raft Trip: $6n + 5 = 29$ - Amusement Park: $14n = 29$ - Balloon Ride: $30n - 40 = 29$ 3. **Solve each equation for $n$: ** **Raft Trip:** $$6n + 5 = 29$$ Subtract 5 from both sides: $$6n + \cancel{5} - \cancel{5} = 29 - 5$$ $$6n = 24$$ Divide both sides by 6: $$\frac{6n}{\cancel{6}} = \frac{24}{\cancel{6}}$$ $$n = 4$$ **Amusement Park:** $$14n = 29$$ Divide both sides by 14: $$\frac{14n}{\cancel{14}} = \frac{29}{\cancel{14}}$$ $$n = \frac{29}{14} \approx 2.07$$ **Balloon Ride:** $$30n - 40 = 29$$ Add 40 to both sides: $$30n - \cancel{40} + \cancel{40} = 29 + 40$$ $$30n = 69$$ Divide both sides by 30: $$\frac{30n}{\cancel{30}} = \frac{69}{\cancel{30}}$$ $$n = \frac{69}{30} = 2.3$$ 4. **Interpret the results:** - Raft Trip: $n=4$ people - Amusement Park: $n \approx 2.07$ people (not a whole number) - Balloon Ride: $n=2.3$ people (not a whole number) Since the number of people must be a whole number, the friends should choose the **Raft Trip** where exactly 4 people can participate for 29 dollars. **Final answer:** The friends should choose the Raft Trip.