Casey ended up with the equation \(5 = 20\), which is a contradiction. This means that there is no solution to the system of equations he created. This indicates that the conditions or relationships between the two landscapers cannot simultaneously satisfy the equations he formed.
Given this scenario, we can analyze the provided options:
-
One landscaper charges $20 for 5 hours of work. - This could still lead to different total costs depending on the fee, but it doesn't imply a contradiction.
-
One landscaper’s hourly rate is $15 lower than the other landscaper’s. - This statement can hold true, but it doesn't necessarily create a contradiction.
-
Both landscapers charge the same hourly rate and the same fee per job. - If this were true, there would be no choice between the two landscapers, and Casey would not need to set up a system of equations.
-
Both landscapers charge the same hourly rate, but not the same fee per job. - This could lead to differing total costs based on how many jobs they do, which could also generate equations that result in a contradiction.
Given that none of the first three options lead to a contradiction where \(5 = 20\), the most logical conclusion is:
Both landscapers charge the same hourly rate and the same fee per job.
Since there is no differentiation between the two landscapers in terms of charges, Casey concludes that there’s no point in hiring one over the other, hence the contradiction in the equations.
3 answers