Let's solve this profit and loss problem step by step. We have students going boating with two different scenarios. In the first scenario, if we add one boat, each boat seats exactly 6 people. Let's visualize this situation.
Now let's examine the second scenario. If we remove one boat, each remaining boat seats exactly 9 people. Notice how we have fewer boats but more people per boat. This gives us another equation to work with in solving our problem.
Now let's set up our mathematical equations. Let n be the original number of boats and x be the total number of students. From scenario 1, we get x equals 6 times n plus 1. From scenario 2, we get x equals 9 times n minus 1. These two equations form our system to solve.
Now let's solve the system by setting the two expressions equal to each other. We get 6 times n plus 1 equals 9 times n minus 1. Expanding both sides gives us 6n plus 6 equals 9n minus 9. Rearranging terms, we get 15 equals 3n, so n equals 5. This means there were originally 5 boats.