At the Adventure Arena amusement park, people can ride the G-Force roller coaster forward or backward. Each train has 15 forward-facing seats and 9 backward-facing seats. Last night, when the lines closed to new riders, 130 people were in line to ride forward and 100 people were in line to ride backward. If the train was completely full for each ride, how many rides did it take for the lines to have the same number of people?

To find out how many rides it took for the lines to have the same number of people, we need to determine when the number of people in line for the forward-facing seats equals the number of people in line for the backward-facing seats. Let's denote: - \( x \) as the number of rides it took for the lines to have the same number of people. We know that each train has 15 forward-facing seats and 9 backward-facing seats. Therefore, the total number of people in line for the forward-facing seats after \( x \) rides is \( 130 + 15x \), and the total number of people in line for the backward-facing seats after \( x \) rides is \( 100 + 9x \). We want to find the value of \( x \) when the number of people in line for the forward-facing seats equals the number of people in line for the backward-facing seats. So, we set up the equation: \[ 130 + 15x = 100 + 9x \] Now, we can solve this equation to find the value of \( x \), which represents the number of rides it took for the lines to have the same number of people. Solve the equation by following steps: - step0: Solve for \(x\): \(130+15x=100+9x\) - step1: Move the variable to the left side: \(130+15x-9x=100\) - step2: Subtract the terms: \(130+6x=100\) - step3: Move the constant to the right side: \(6x=100-130\) - step4: Subtract the numbers: \(6x=-30\) - step5: Divide both sides: \(\frac{6x}{6}=\frac{-30}{6}\) - step6: Divide the numbers: \(x=-5\) The solution to the equation is \( x = -5 \). However, the number of rides cannot be negative. This means that the lines will have the same number of people after 5 rides.