Let's start by defining the variables we need for the problem:
$w$: the number of weeks you'll be renting the computer.
$C_1$: the total cost of the first option.
$C_2$: the total cost of the second option.
According to the problem statement, the total cost $C_1$ of the first option is given by a base fee of $280 plus $15 per week:
$$C_1 = 280 + 15w$$
The total cost $C_2$ of the second option is given by a flat fee of $22 per week:
$$C_2 = 22w$$
We want to find out after how many weeks $w$ the first option becomes the better deal. In other words, we want to find the point where $C_1 < C_2$:
$$280 + 15w < 22w$$
Simplifying this inequality, we get:
$$280 < 7w$$
$$w > \frac{280}{7}$$
$$w > 40$$
Therefore, after 40 weeks, the first option becomes the better deal, since the total cost will be lower than the flat rate option.