Question

Use the given cost table for the same product from two different companies to create a linear system.

Answer 1

Answer:

Step-by-step explanation:

Calculate the slope for both companies (1 and 2) by picking two points from the table for each company and then determine the "Rise/Run."  

The costs for both companies:

                 Cost ($)      

Shirts         1 f(x)       2 g(x)

  5           117.75       51.75

15          159.25    115.25

Rise            41.5       51.75

Run            10          10    

Rise/Run    4.15         6.35

The slopes of these lines are 4.15 and 6.35

Comp. 1:  y = 4.15x + b

Comp. 2:  y = 6.35x + b

Strangely, both equations have a b that is not zero.  This means that, mathematically, there is a cost for doing 0 shirts.  Only in America.

One can see this issue by looking at the difference in cost for the first 5 shirts compared to the second set of 5.  For Company 1, the first 5 is $117.75, but the second 5 brings us to only a total of $138.50 - a difference of only $20.75.  Same with Company 2.  The first 5 shirts costs much more than the second five.  The third set of 5 shirts (10 - 15) is consistent with the increase with the second set (5-10).

While this doesn't make sense, we must assume there is an initial starting charge, which will be the value of b in the y=mx+b format.

Top calculate b in both equations, enter one of the data points (e.g., (5,117.8) for company 1, and (5,51.75) for company 2 and solve each equation for b.

The results will given these equations:

Company 1:   f(x) = 4.15x + 97,   and

Company 2:  g(x) = 6.35x + 20

Take one shirt into company 1 and you're out $97 before paying for the one laundered shirt.  $20 for Company 2.

But each company has a different shirt price:

Company 1:  $4.15/shirt

Company 2:  $6.35/shirt

We can find the point at which both companies charge the same for x number of shirts by setting the equations equal to each other [i.e., their costs are the same] and solving for x:

4.15x + + 97 = 6.35x + 20

x = 35 shirts

At 35 shirts, both companies will charge $242.30

We can also graph the two equations to find their intersection (both have the same cost).  The graph also highlights the absurdity of interpreting anything under 1 shirt.  So the range of the functions is x ≥ 1.

See attachment.