Math Discussion 1.1

In the overview of this module, we saw how a linear equation can be applicable to the grocery store expenditure. Think about 3 different examples where you can formulate 3 different linear equations. Then, share those examples and linear equations. Include the x-intercept and y-intercept of each of your linear equations. Comment on the validity of the other students’ linear equations. None of your linear equations must have expenditure as the variable. (Think beyond the grocery store example.) Example from module description: Suppose that you go to a grocery store to buy some cola bottles. Your expenditure depends on how much you spend on gas and how many bottles you buy at the grocery store. The amount you spend on gas does not depend on how many bottles you buy, but the amount you spend in the grocery store does. Hence, your gas expenditure can be considered fixed cost, and your cola bottles’ expenditure can be considered variable cost. Now, suppose also that your gas expenditure for each trip to the grocery store is $3, and the selling price of each cola bottle is $2. If your total expenditure is expressed as y, and the number of cola bottles is expressed as x, then one can say that y = 3 + 2x.



Example 1. Suppose it is girl scout cookie season. :) You have committed to buy ten boxes of cookies from your own daughter, and also to buy three boxes from every other girl scout that comes to your door selling cookies. The total amount of cookies you buy does not depend on how many boxes you buy from each girl, but on the number of girls that come to your door. Therefore, the cookies you buy from your own daughter can be considered fixed, and the cookies from other girl scouts can be considered variable. If your total cookie purchase (in boxes) is expressed as y, and the number of girl scouts you buy cookies from is expressed as x, then y = 10 + 3x.

• Solving the x-intercept for y = 10 + 3x (y = 0)
  
  0 = 10 + 3x
  
  -10 = 3x
  
  -(10/3) = x
  
  -3.33 = x
  
  x-intercept is (-3.33, 0)


• Solving the y-intercept for y = 10 + 3x (x = 0)
  
  y = 10 + 3*0
  
  y = 10 + 0
  
  y = 10
  
  y-intercept is (0, 10)

Example 2. Suppose that you are considering carpooling to reduce your 20 mile per day commute to two miles per day to a carpool meetup spot, plus the once or twice weekly full commute when it's your turn to drive (based on a five day work week). The total number of miles you drive per week does not depend on how many miles in your commute, but on the frequency you have to make that drive. Therefore, the mileage to your carpool meetup is fixed, and the mileage for the rest of the week is variable. If your total weekly mileage is expressed as y, and the frequency you drive the carpool all the way is expressed as x, then y = 2*5 + 20x.

• Solving the x-intercept for y = 2*5 + 20x (y = 0)
  
  0 = 2*5 + 20x
  
  0 = 10 + 20x
  
  -10 = 20x
  
  -(10/20) = x
  
  -0.5 = x
  
  x-intercept is (-0.5, 0)


• Solving the y-intercept for y = 2*5 + 20x (x = 0)
  
  y = 2*5 + 20*0
  
  y = 10 + 0
  
  y = 10
  
  y-intercept is (0, 10)

Example 3. Suppose that you are enrolled in an eleven-week graduate class, and are scheduled to complete ten homework assignments, twenty online discussions, and one final exam (31 total assignments). Suppose also that you alot two hours for easy assignments and four hours for difficult assignments. The total number of hours you need to plan for your schoolwork does not depend on the number of assignments, but on how many of them are easy and how many are difficult. If the total number of hours you need to expend on your assignments is expressed as y, and the number of assignments that are difficult is expressed as x, then y = 2(31-x) + 4x.

• Solving the x-intercept for y = 2(31-x) + 4x (y = 0)
  
  0 = 2(31-x) + 4x
  
  0 = 2*31 - 2x + 4x
  
  0 = 62 + 2x
  
  -62 = 2x
  
  -(62/2) = x
  
  -31 = x
  
  x-intercept is (-31, 0)


• Solving the y-intercept for y = 2(31-x) + 4x (x = 0)
  
  y = 2(31-0) + 4*0
  
  y = 2*31 + 0
  
  y = 62
  
  y-intercept is (0, 62)