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Jessica is selling books during the summer to earn money for college. She earns a commission on each sale but has to pay for her own expenses.

After a month of driving from neighborhood to neighborhood and walking door-to-door, she figures out that her weekly earnings are approximately a linear function of the number of doors she knocks on.

She writes the equation of the function like this: [tex]E(x) = 7x - 25[/tex], where [tex]x[/tex] is the number of doors she knocks on during the week and [tex]E(x)[/tex] is her earnings for the week in dollars.

Which of the following are reasonable interpretations of the [tex]y[/tex]-intercept of Jessica's function? Check all that apply.

A. Her expenses are [tex]\$25[/tex] per week.
B. She can earn [tex]\$7[/tex] per week even if she does not knock on any doors.
C. If she does not knock on any doors at all during the week, she will lose [tex]\$25[/tex].
D. She will lose [tex]\$7[/tex] per week if she does not knock on any doors.



Answer :

GinnyAnswer
To analyze Jessica's weekly earnings function [tex]\( E(x) = 7x - 25 \)[/tex], let's examine each interpretation of the [tex]$y$[/tex]-intercept step-by-step.

### Understanding the Equation
- E(x) represents Jessica’s weekly earnings in dollars.
- x represents the number of doors she knocks on during the week.
- The equation is [tex]\( E(x) = 7x - 25 \)[/tex].

### Interpreting the y-intercept
The y-intercept of the function, which is the value of [tex]\( E(x) \)[/tex] when [tex]\( x = 0 \)[/tex], is [tex]\( -25 \)[/tex]. This value [tex]\( -25 \)[/tex] provides insight into Jessica's fixed costs or fixed losses that do not depend on the number of doors she knocks on.

Now let's evaluate each interpretation:

#### Interpretation A: Her expenses are \[tex]$25 per week. - When \( x = 0 \), the earnings are \( E(0) = 7(0) - 25 = -25 \). This means Jessica has a negative earning of \$[/tex]25 when she doesn't knock on any doors, which can be interpreted as her weekly expenses. Therefore, this interpretation is true.

#### Interpretation B: She can earn \[tex]$7 per week even if she does not knock on any doors. - When \( x = 0 \), her earnings are \( -25 \). This means she does not earn anything if she doesn't knock on any doors; instead, she incurs a loss of \$[/tex]25. Thus, this interpretation is false.

#### Interpretation C: If she does not knock on any doors at all during the week, she will lose \[tex]$25. - Again, when \( x = 0 \), \( E(0) = -25 \). This directly means that Jessica incurs a loss of \$[/tex]25 if she does not knock on any doors. This interpretation is true.

#### Interpretation D: She will lose \[tex]$7 per week if she does not knock on any doors. - This interpretation is incorrect because the loss when she does not knock on any doors is \$[/tex]25, not \$7. So, this interpretation is false.

Combining our evaluations:

- A. True
- B. False
- C. True
- D. False

The final result, expressed as a tuple where 1 represents true and 0 represents false, is:
```
(1, 0, 1, 0)
```

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