Let's define the variables:

• x: the number of months Noah uses the service.
• y: the total cost of the service.

Now, let's break down the pricing for each company:

Company A:

• One-time fee: $10.
• Monthly cost: $6 per month.

Thus, the total cost for Company A after x months is: $y = 10 + 6x$

Company B:

• One-time fee: $20.
• Monthly cost: $3 per month.

Thus, the total cost for Company B after x months is: $y = 20 + 3x$

System of Equations:

The system of equations representing this situation is: $y = 10 + 6x$ $y = 20 + 3x$

This system shows how the total cost for each company depends on the number of months Noah uses the service.

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Would you like more details or explanations? Here are 5 related questions to explore:

1. How can you determine which company would be cheaper after a certain number of months?
2. If Noah uses the service for 5 months, what would be the total cost for each company?
3. How can you find the month when the total cost for both companies will be the same?
4. What happens to the total cost if Noah uses the service for a very long time (e.g., 100 months)?
5. How can you graph these two equations on a coordinate plane?

Tip: To determine when both companies cost the same, set the two equations equal to each other and solve for $x$.