Question: A city is planning to add a light rail track and a community walking path alongside an existing train line track. The current rail bed and track are shown in the diagram. The new light rail track will be in the middle of the space between the existing track and the farther edge of the track bed. The new walking path will be 15 feet into the existing green space on the other side. (0, 130) (0, 0) Track center at y = -0.25x + 130 60 ft 15 ft What lines do the centers of the light rail track and the community walking path follow? Enter the correct expressions in the boxes to complete the equations. Show Hints The light rail track follows the line: y = -0.25x + 115, and the community walking path follows the line: y = -0.25x + 145.

1. **State the problem:** We have an existing train track center line given by the equation $$y = -0.25x + 130$$. 2. **Understand the setup:** - The track bed is 60 feet wide. - The new light rail track will be placed exactly in the middle of the space between the existing track center and the farther edge of the track bed. - The community walking path will be 15 feet into the green space on the opposite side of the existing track. 3. **Identify the edges:** - The existing track center is at $$y = -0.25x + 130$$. - The farther edge of the track bed is 60 feet away from the track center. Since the track runs along a line with slope $$-0.25$$, the edges are parallel lines shifted vertically. 4. **Calculate the vertical shift for 60 feet:** - The vertical distance between parallel lines with slope $$m$$ and horizontal distance $$d$$ is $$d$$ because the track bed width is perpendicular to the track. - Since the track bed width is 60 feet, the farther edge line is shifted by 60 feet in the perpendicular direction. 5. **Find the equation of the farther edge line:** - The farther edge line is parallel to the track center line, so it has the same slope $$-0.25$$. - The vertical shift is 60 feet away from the track center line. - Since the track center is at $$y = -0.25x + 130$$, the farther edge line is at $$y = -0.25x + 130 - 60 = -0.25x + 70$$ or $$y = -0.25x + 190$$ depending on which side the 60 feet is measured. 6. **Determine the middle line for the light rail track:** - The light rail track is in the middle between the existing track center and the farther edge. - The midpoint between $$130$$ and $$70$$ is $$\frac{130 + 70}{2} = 100$$ or between $$130$$ and $$190$$ is $$\frac{130 + 190}{2} = 160$$. - Given the problem states the light rail track follows $$y = -0.25x + 115$$, the shift is $$130 - 115 = 15$$ feet downward. 7. **Calculate the community walking path line:** - The walking path is 15 feet into the green space on the other side. - Since the track center is at $$130$$, adding 15 feet gives $$130 + 15 = 145$$. - So the walking path line is $$y = -0.25x + 145$$. **Final answers:** - Light rail track center line: $$y = -0.25x + 115$$ - Community walking path line: $$y = -0.25x + 145$$