1. **State the problem:** We have two families buying tickets for children and adults at Rollercoaster World. The Brown family paid 170 for 3 children and 2 adults, and the Peckham family paid 360 for 4 children and 6 adults. We want to plot the system of equations representing these purchases. 2. **Write the equations:** Let $x$ be the price of a child's ticket and $y$ be the price of an adult's ticket. The Brown family: $$3x + 2y = 170$$ The Peckham family: $$4x + 6y = 360$$ 3. **Find intercepts for each equation:** - For the Brown family equation: - $x$-intercept (set $y=0$): $$3x = 170 \Rightarrow x = \frac{170}{3} \approx 56.67$$ - $y$-intercept (set $x=0$): $$2y = 170 \Rightarrow y = 85$$ - For the Peckham family equation: - $x$-intercept (set $y=0$): $$4x = 360 \Rightarrow x = 90$$ - $y$-intercept (set $x=0$): $$6y = 360 \Rightarrow y = 60$$ 4. **Find the intersection point of the two lines:** Solve the system: $$3x + 2y = 170$$ $$4x + 6y = 360$$ Multiply the first equation by 3: $$9x + 6y = 510$$ Subtract the second equation: $$(9x + 6y) - (4x + 6y) = 510 - 360$$ $$5x = 150 \Rightarrow x = 30$$ Substitute $x=30$ into the first equation: $$3(30) + 2y = 170 \Rightarrow 90 + 2y = 170 \Rightarrow 2y = 80 \Rightarrow y = 40$$ 5. **Summary of points for plotting:** - Brown family line: intercepts at $(56,0)$ and $(0,85)$ (rounded to nearest integers) - Peckham family line: intercepts at $(90,0)$ and $(0,60)$ - Intersection point: $(30,40)$ These points can be plotted on a graph to visualize the ticket prices. **Final answer:** - Brown family line: passes through $(56,0)$ and $(0,85)$ - Peckham family line: passes through $(90,0)$ and $(0,60)$ - Intersection point: $(30,40)$ which represents the price of a child's ticket as 30 and an adult's ticket as 40.