Systems of equations are used in many cost models in the real world.

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Systems of equations are used in many cost models in the real world. Here is an example scenario for you to consider.
• Suppose you are running a carnival. You are selling hamburgers and sodas. A hamburger is $1.75 and a soda is .75.
• You expect to make a total of $117.50 for the day
• You also plan to sell 120 hamburgers and sodas
• How many sodas and hamburgers will you sell each?
• Suppose you decide to change the price ratio between hamburgers and soda’s so that they produce a more equal consumption of hamburgers and sodas.
o What would you change the price to for each and why?
o What would be the new amount of hamburgers and soda each at your price points?

 

Sample  Solution

To determine the number of sodas and hamburgers to be sold, we can set up a system of equations:

Let:

  • x represent the number of hamburgers sold.
  • y represent the number of sodas sold.

Equations:

  1. Total revenue: 1.75x + 0.75y = 117.50
  2. Total items sold: x + y = 120

Solving the system of equations:

  • Solve the second equation for x: x = 120 – y
  • Substitute this expression for x in the first equation: 1.75(120 – y) + 0.75y = 117.50
  • Simplify and solve for y: 210 – 1.75y + 0.75y = 117.50 0.75y = 92.50 y = 123.33 (rounded to two decimal places)
  • Substitute y back into the equation x = 120 – y: x = 120 – 123.33 = -3.33 (rounded to two decimal places)

Interpreting the results:

  • The negative value for x indicates that the given prices and sales goal are inconsistent. It’s not possible to sell a negative number of hamburgers.

Adjusting prices for more equal consumption:

  • To encourage more equal consumption, we can reduce the price of hamburgers and increase the price of sodas.
  • New prices:
    • Hamburgers: $1.50
    • Sodas: $1.00

New system of equations:

  1. 1.50x + 1.00y = 117.50
  2. x + y = 120

Solving for the new quantities:

  • Solve the second equation for x: x = 120 – y
  • Substitute into the first equation: 1.50(120 – y) + 1.00y = 117.50 180 – 1.50y + 1.00y = 117.50 0.50y = 62.50 y = 125
  • Substitute y back into x = 120 – y: x = 120 – 125 = -5

New results:

  • Hamburgers: 55 (rounded up from -5, as we cannot sell a negative number)
  • Sodas: 65

Therefore, with the adjusted prices, you would sell approximately 55 hamburgers and 65 sodas to achieve a more equal consumption and meet your sales goal.

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