In this problem, we need to compare the cost of renting a rototiller from two different companies. At Tools Plus, there's a one-time fee of $20 plus an hourly rate of $10. At Tools Unlimited, there's only an hourly fee of $15 with no additional charges. We need to find when these two rental options would cost exactly the same amount. Let's represent this situation with a system of equations, where x is the number of hours and y is the total cost.
For Tools Plus, the equation is y equals 10x plus 20, where the one-time fee of $20 is represented by the constant term. For Tools Unlimited, the equation is simply y equals 15x. To find when these two rental options cost the same, we set these equations equal to each other. This gives us 10x plus 20 equals 15x. The intersection point of these two lines represents the solution to our system of equations.
Now let's solve the system of equations. We have 10x plus 20 equals 15x. Subtracting 15x from both sides, we get 10x minus 15x equals negative 20, which simplifies to negative 5x equals negative 20. Dividing both sides by negative 5, we get x equals 4. This means that after 4 hours, the rental cost will be the same at both places. To find this cost, we substitute x equals 4 into either equation. Using the Tools Plus equation, y equals 10 times 4 plus 20, which gives us y equals 40 plus 20, or 60 dollars. So after 4 hours, both rental options will cost 60 dollars.
Let's compare the costs at different rental durations. For 1 hour, Tools Plus costs $30 while Tools Unlimited costs only $15. For 2 hours, Tools Plus costs $40 and Tools Unlimited costs $30. For 3 hours, the costs are $50 and $45 respectively. At exactly 4 hours, both companies charge $60. After that, the trend reverses. For 5 hours, Tools Plus charges $70 while Tools Unlimited charges $75. For 6 hours, the costs are $80 and $90 respectively. We can see that before 4 hours, Tools Unlimited is the cheaper option. At exactly 4 hours, both options cost the same at $60. After 4 hours, Tools Plus becomes the more economical choice. Therefore, our answer is the point (4, 60).
Let's summarize what we've learned. We compared the rental costs between Tools Plus, which charges a one-time fee of $20 plus $10 per hour, and Tools Unlimited, which charges $15 per hour with no additional fees. We set up a system of equations where y equals 10x plus 20 for Tools Plus, and y equals 15x for Tools Unlimited. To find when these costs are equal, we set the equations equal to each other: 10x plus 20 equals 15x. Solving for x, we got x equals 4 hours. By substituting this value back into either equation, we found that at exactly 4 hours, both rental options cost $60. Before 4 hours, Tools Unlimited is cheaper, and after 4 hours, Tools Plus becomes the more economical choice.