A hiker completed a $20$-mile trail in $7$ hours. For the first part of the trail, the hiker walked at an average speed of $3$ mph. For the second part, the hiker walked at an average speed of $2$ mph due to rough terrain. If $x$ is the time, in hours, spent on the first part of the trail and $y$ is the time, in hours, spent on the second part, which system of equations represents this situation?