Linear Systems Theory HW

Question

I have this as homework in an integrated three math class. We are working on linear systems (thanks for the help in advance!!!):

Bob purchases the same four items every week (apples, ice cream, cans of soup, and peanut butter), but he refuses to tell us how much each of them costs. (Maybe he’s embarrassed). He will, for some reason, tell us how many of each thing he bought, and the total bill for that week.

 

a. Assuming he buys each item each week, how many weeks would it take you to figure out how much each item costs? Why would it take that long to figure out?

b. Would any additional weeks of data make it easier or harder to figure out how much each item costs?

c. Would it help if he didn’t purchase some of the items some week? Would that allow you to figure out how much each thing cost in fewer weeks?

d. Could Bob make his purchases in such a way that you couldn’t ever figure out how much each item was? How might he do this?

Answer 1

This looks like a system of equations, where we have a certain number of unknowns.  The number of related equations must be greater than or equal to the number of unknowns in order to solve for the unknowns.

 

Let a = the number of apples

Let b = the number of ice cream

Let c = the number of can soup

Let d = the number of peanut butter

 

According to your question, these are known.

 

 

Let C₁ = cost of apples

Let C₂ = cost of ice cream

Let C₃ = cost of can soup

Let C₄ = cost of peanut butter

 

Using these variables, we can set up equations.

 

 

The first equation will represent the total number of items for 1 week.

 

a + b + c + d = T

 

 

The second equation will represent the total cost of each item for 1 week.

 

C₁a + C₂b + C₃c + C₄d = C_T

 

 

The third equation will represent the total cost of each item for 2 weeks.

 

2(C₁a + C₂b + C₃c + C₄d) = 2C_T

 

 

The fourth equation represents the cost of each item for 3 weeks.

 

3(C₁a + C₂b + C₃c + C₄d) = 3C_T

 

 

Because we have 4 variables, we will need 4 equations.  We already established those 4 equations.  The last equation represents the total cost of items for 3 weeks.  Therefore, we can say it will take at least 3 weeks to find the cost of each item.

 

 

b)

 

Any additional weeks will make it difficult to solve for the cost of each item.  In a system of equations, solving up to 3 equations is simple.  But if we increase the number of equations and variables, then solving it by hand will be difficult, unless one of the equations does not contain all the variables, or have a computer program that will allow you to easily solve 5 or 10 equations.

 

 

c)

 

If he did not purchase some of the items per week, then we will only have 1 or 2 less variables and equations to work with.  As I mentioned before, the less variables and equations you have to work with, the easier it is to solve.

 

 

d)

 

Yes he can.  He would not tell you how much of each item he bought and their cost.  Without those two pieces of information, writing some equations will be impossible.

 

Answer 2

 

I have this as homework in an integrated three math class. We are working on linear systems (thanks for the help in advance!!!):
Bob purchases the same four items every week (apples, ice cream, cans of soup, and peanut butter), but he refuses to tell us how much each of them costs. (Maybe he’s embarrassed). He will, for some reason, tell us how many of each thing he bought, and the total bill for that week.

a. Assuming he buys each item each week, how many weeks would it take you to figure out how much each item costs? Why would it take that long to figure out?

At least 4 weeks. You need 4 linearly independent equations to solve the four unknowns.

b. Would any additional weeks of data make it easier or harder to figure out how much each item costs?

May or may not. If different number of items were purchased in each of the first four weeks, then only four weeks would be required, and additional weeks of data would not be necessary.

c. Would it help if he didn’t purchase some of the items some week? Would that allow you to figure out how much each thing cost in fewer weeks?

It would help to determine the cost of an item faster, but not all four items

d. Could Bob make his purchases in such a way that you couldn’t ever figure out how much each item was? How might he do this?

Buy the same number (or multiple of the same number) of items every week.