Let's analyze and compare the two carriers based on the provided information.
### Carrier A
We are given a table with the total costs for Carrier A over different months. To determine the monthly cost and the initial cost (intercept), we can use the given data points.
| Months (x) | Total Cost (y) |
|------------|----------------|
| 1 | 70 |
| 2 | 115 |
| 3 | 160 |
| 4 | 205 |
| 5 | 250 |
From the table, we can see:
- At 1 month: the total cost is 70 dollars.
- At 2 months: the total cost is 115 dollars.
- At 3 months: the total cost is 160 dollars.
- At 4 months: the total cost is 205 dollars.
- At 5 months: the total cost is 250 dollars.
By comparing the costs for consecutive months:
[tex]\[ \text{Cost difference between any two consecutive months} = \$115 - \$70 = \$45 \][/tex]
Thus, the cost per month (slope) for Carrier A is [tex]$45.
Now, to find the initial cost (intercept) for Carrier A, we can use the first data point (1 month, 70 dollars):
\[ \text{Initial cost} = 70 - 45 \times 1 = 25 \]
So, the equation for the total cost \( y \) of the phone and \( x \) months of service at Carrier A is:
\[ y = 45x + 25 \]
### Carrier B
The total cost \( y \) of the phone and \( x \) months of service at Carrier B is given by the equation:
\[ y = 55x + 300 \]
### Comparing Monthly Costs
- Carrier A: \( y = 45x + 25 \) (slope is 45)
- Carrier B: \( y = 55x + 300 \) (slope is 55)
The slope represents the monthly cost for each carrier. Comparing the slopes, Carrier A charges \( \$[/tex]45 \) per month, while Carrier B charges [tex]\( \$55 \)[/tex] per month. Therefore, Carrier A charges less per month.
### Finding the Break-Even Point
To find the point where the total costs are the same for both carriers, we set the two equations equal to each other:
[tex]\[ 45x + 25 = 55x + 300 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
45x + 25 = 55x + 300
\][/tex]
[tex]\[
25 - 300 = 55x - 45x
\][/tex]
[tex]\[
-275 = 10x
\][/tex]
[tex]\[
x = \frac{-275}{10} = -27.5
\][/tex]
Therefore, the total costs for both carriers would be the same after [tex]\(-27.5\)[/tex] months of service. This negative value indicates that, in practical terms, the two carriers will never have the same total cost within the context of real-world months of service.
