Making Real-world Connections with Linear Systems – Solving Graphically

First let’s solve a system using an algebraic method and then graphically…

1.  For 1980 through 1990, the coal production in the world, C (in millions of metric tons), and crude petroleum production, P (in millions of metric tons), can be modeled by the following equations. Let t = 0 represent 1980. During which year were the production levels equal?

  1. Solve the system algebraically (use substitution…set them equal to each other). Show your work below.
  1. Using your calculator, enter the two equations into Y1 and Y2. Set your window as shown below. Find the intersection point by pressing 2nd Trace (Intersection). Press enter 3 times to get the intersection point.
  1. What is the solution to the system? ______
  2. During which year were the production levels equal? Recall t = 0 represent 1980. ______

Now let’s solve using a linear regression…

2.  The table below shows the winning times for the Olympic 400-meter dash from 1968 to 2000 for both men and women. According to the data, approximately when will the women’s winning times be equal to the men’s winning times? Let represent 1968.

Year / 1968 / 1972 / 1976 / 1980 / 1984 / 1988 / 1992 / 1996 / 2000
Men’s Time / 43.68 / 44.66 / 44.26 / 44.60 / 44.27 / 43.87 / 43.50 / 43.49 / 43.84
Women’s Time / 52.03 / 51.08 / 49.29 / 48.88 / 48.83 / 48.65 / 48.83 / 48.25 / 49.11
  1. Enter the data into the following lists: L1—years since 1968, L2—men’s time, L3—women’s time
  2. Be sure to turn on both STATPLOTs. What do you press to tell the calculator to set a window according to your data points? ______
  3. Find each linear regression. Then enter each linear regression into Y1 and Y2.

Linear regression #1: ______Linear regression #2: ______

  1. Find the intersection point. What is the solution to the system? ______
  2. When will the women’s winning times be equal to the men’s winning times? ______

3.  The table right shows the consumption of broccoli and cauliflower (lb/person) in the US from 1970 to 2000. According to the data, when will the consumption of broccoli be equal to the consumption of cauliflower? What will be the consumption of broccoli and cauliflower? Let represent 1970.

Year / Broccoli
(lb/person) / Cauliflower
(lb/person
1970 / 0.5 / 2.6
1975 / 1.0 / 2.6
1980 / 1.4 / 3.6
1985 / 2.6 / 4.0
1990 / 3.4 / 4.3
1995 / 4.4 / 5.2
2000 / 5.7 / 5.8
  1. Enter the data into the following lists: L1—years since 1970, L2—broccoli consumption, L3—cauliflower consumption
  2. Be sure to turn on both STATPLOTs. What do you press to tell the calculator to set a window according to your data points?
  3. Find each linear regression. Then enter each linear regression into Y1 and Y2.

Linear regression #1: ______

Linear regression #2: ______

  1. Find the intersection point. What is the solution to the system? ______
  2. When will the women’s winning times be equal to the men’s winning times? ______

Got it?

4.  The table shows the population of the San Diego and Detroit metropolitan regions. When were the populations of these regions equal? What was the population?

  1. Circle the first calculator step in solving the problem.

Calculate the intersection enter the data into lists enter Y1

  1. Write the name of the list you will use (L1, L2, or L3) next to the data type.

_____ population of San Diego _____population of Detroit _____years since 1950

  1. Write the linear regressions.

Linear regression #1: ______Linear regression #2: ______

  1. Circle the correct word to complete the sentence.
  2. What is the intersection point? ______
  3. The population of San Diego and Detroit were equal sometime during the year ______.
  4. The population was about ______.

Focus Question—What is the significance of the break-even point and how can it be used?

Break-Even Analysis

Break-even analysis is used in business and finance to help companies make wise decisions. Part of doing a break-even analysis is finding the break-even point. As its name implies, it is the point at which a balance is reached between expenses and income, between cost and revenue.

In business, the value of the break-even point represents when the cost invested equals the amount returned (the revenue). In other words, there is neither profit nor loss. Profit is made when there is more revenue than cost or when a business makes more than it spends. Loss occurs when the business spends more than it makes.

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  1. What is the break-even point?
  2. When is profit made?

Example of a Break-even Point

Suppose a school is selling candy for a fundraiser. Assume that it costs the school $1 for every bar of candy it purchases and that each bar is going to be sold for $2. Further assume that the school has spent $100 in advertising for this fundraiser.

How many candy bars must the school purchase and sell before it breaks even? In other words, what is the break-even point (the point at which the school has neither lost money nor made a profit)?

Let x = the number of packs of candy bought and sold at any particular time.

Let y = the cost/income in dollars

3.  Write an equation to represent the cost of the candy: ______

4.  Write an equation to represent the income from selling the candy: ______

5.  When does cost equal income?

6.  What is the solution?

7.  How many candy bars must the school sell to break even?

8.  Complete the sentence: If the school sells less than ______candy bars they will ______money, but if the school sells more than ______candy bars, they will make a ______.

Example 2

Suppose your business makes high-grade PDAs, personal digital assistants. It costs you $80,000 a month for fixed costs, such as rent, employee salaries, and other essential items. Also, it costs $300 to make each PDA, called a unit. If you are able to sell each unit for $500, how many PDAs do you need to sell to break even?