The cost of a fill on acrylic nails is a function of how many broken nails. MArtha is charged $34 for 5 broken nails, while Grace is charged $26 for 1 broken nail.

a)express the total cost as a ntion of the number of broken nails.
b)What is the slpe of this ntion and what does it represent?
c)Another salon charges $6 per nail for a fill regardlesss of the number broken nails. How many nails would have to be broken to make it more economical to go to Martha and Garce's salon?

cost=A*number nails + fixed cost

Can you take it from here?
The easy way is to notice A is the slope, or 8dollars/4nails or 2dollars/nail.

a) To express the total cost as a function of the number of broken nails, we need to determine the relationship between the number of broken nails and the cost. From the given information, we have two data points: $34 for 5 broken nails and $26 for 1 broken nail.

Let's assume the number of broken nails is represented by 'x' and the total cost is represented by 'C(x)'. We can create two equations using the given data:

Equation 1: C(5) = $34
Equation 2: C(1) = $26

We can now find the equation of the line that represents this relationship. We will use the slope-intercept form of a linear equation: y = mx + b, where y represents the total cost, x represents the number of broken nails, m represents the slope, and b represents the y-intercept.

Using Equation 1:
C(5) = $34
m(5) + b = $34
5m + b = $34

Using Equation 2:
C(1) = $26
m(1) + b = $26
m + b = $26

We now have a system of equations:
5m + b = $34
m + b = $26

We can solve this system of equations to find the values of m and b. Subtracting the second equation from the first equation, we get:
(5m + b) - (m + b) = $34 - $26
5m - m = $34 - $26
4m = $8
m = $8/4
m = $2

Now that we have the slope, we can substitute it back into one of the equations to find the value of b. Using Equation 2:
m + b = $26
$2 + b = $26
b = $26 - $2
b = $24

Therefore, the function for the total cost, C(x), is:
C(x) = $2x + $24

b) The slope of this function, which is 2, represents the cost of each additional broken nail. It tells us that for each additional broken nail, the cost increases by $2. The slope indicates the rate of change of the total cost with respect to the number of broken nails.

c) Let's compare the cost of the other salon, which charges $6 per nail, with the cost of Martha and Grace's salon. We need to find the number of broken nails, denoted by 'n', at which the cost of Martha and Grace's salon becomes more economical.

To find this point, we need to set the two costs equal to each other:
$6n = $2n + $24

Simplifying the equation:
$6n - $2n = $24
$4n = $24
n = $24/$4
n = 6

Therefore, if Martha and Grace's salon charges $2 per broken nail, it would be more economical to go to their salon if 6 or more nails are broken.