Day 81 - Systems of Equations Application

Goals: Students will be able to solve real world problems by using systems of equations. Students will be able differentiate between two types, break even and starting value problems and use the appropriate strategy to solve each.

Two Types of Items These problems compare to different types of items. The x and y variables represent each of the items. Equation 1 represents the total cost of all items (Remember that the # of times * cost = the total cost) Equation 2 represents information about the number of items.

Example Juan wants to buy his friend balloons for her birthday. Foil balloons cost $3.75 each and plain balloons cost $1.75 each. Juan can spend $25.50 on the balloons. If he buys ten balloons, how many of each type can he buy? X=foil balloons Y=plain balloons 3.75x + 1.75y = 25.50 x + y = 10

Example 3.75x + 1.75y = 25.50 x + y=10 1.75y = -3.75x+25.5

Example (Zoom #6)

Example (Trace)

Example Juan can buy four foil and six plain balloons.

Break Even Problems These problems compare cost and profits. x=number of items or people y=profit or cost Equation one represents total expenses ($+$=y) Equation two represents profit ($x=y)

Example Marcus and Maria are having a fundraiser for their daughter’s political campaign. They have spent $3000 in set-up cost and each guest will cost another $20 each for food. Each guest purchases a ticket to the event for $75. How many people must come for them to break even?

Example y=(3000+20x) y=75x

Example (Zoom #6)

Example (Set window)

Example (Trace) 55 people must buy tickets for the fundraiser to break even.

Starting Value Problems These problems determine which of two products will be of the greatest value. x=independent variable y=dependant variable Equation one represents y=starting value1+rate of change1 * x Equation two represents y=y=starting value2+rate of change2 * x

Example The homeowners association is voting on a new policy. Option one will charge each owner $100 plus $5 per square foot of their home. Option B is to charge each owner $175 plus $1.50 per square foot. Option A: y=5x+100 Option B: y=1.5x+175

Example At how many square feet will each plan cost the same amount? How much will that be? 21.4 square feet will cost $207.14 under either plan.

Example Which plan would be the better choice? Since most bedrooms are larger than 21 square feet the homeowners will save money with option B.