Answer:
<h3>a. Compare the y-intercepts and rates of changes.</h3>
Remember that in the form
, the coeffiecient of the variable is the rate of change, and the constant is the y-intercept.
In this case, we have the equation
![p=5b-15](https://tex.z-dn.net/?f=p%3D5b-15)
Where
is the profit and
is the number of bracelets. Her rate of change is $5 per bracelet and its y-intercept is at -15.
On the other hand, the table below represents Kate's profits
Bracelets sold (x) Profit (in dollars) (y)
1 5
2 10
3 15
4 20
First, we need to find the rate using the following formula and two pairs of the table (1,5) and (3,15)
![r=\frac{y_{2}-y_{1} }{x_{2}-x_{1} }\\ r=\frac{15-5}{3-1}=\frac{10}{2}\\ r=5](https://tex.z-dn.net/?f=r%3D%5Cfrac%7By_%7B2%7D-y_%7B1%7D%20%20%7D%7Bx_%7B2%7D-x_%7B1%7D%20%7D%5C%5C%20r%3D%5Cfrac%7B15-5%7D%7B3-1%7D%3D%5Cfrac%7B10%7D%7B2%7D%5C%5C%20%20r%3D5)
The are of Kate's profist is 5 dollars per bracelet.
Now we use the point-slope formula
![y-y_{1} =m(x-x_{1} )\\y-5=5(x-1)\\y=5x-5+5\\y=5x](https://tex.z-dn.net/?f=y-y_%7B1%7D%20%3Dm%28x-x_%7B1%7D%20%29%5C%5Cy-5%3D5%28x-1%29%5C%5Cy%3D5x-5%2B5%5C%5Cy%3D5x)
This means the y-intercept is at zero.
If we compare, we would deduct that Carol's profit begins with -$15, while Kate's profit begins at $0, in other words, Carol has a debt. Also, both of them have the same rate of profit $5 per bracelet.
<h3>b. How much will each girl make if she sells 30 bracelets.</h3>
Carol would make
![p=5b-15=5(30)-15\\p=135](https://tex.z-dn.net/?f=p%3D5b-15%3D5%2830%29-15%5C%5Cp%3D135)
$135 profit.
Kate would make
![y=5x=5(30)=150](https://tex.z-dn.net/?f=y%3D5x%3D5%2830%29%3D150)
$150 profit.
So, Kate would make $15 more than Carol.