In this sub-unit, students will be introduced to the concept of systems of linear equations and their solutions. They will learn to solve these systems using accurate graphing, substitution, and linear combination. The concept will be introduced in the context of a business model (cost and revenue curves) and the need to determine how many units of a product will need to be sold before the business becomes profitable. Students will then conduct their own research into cost and revenue and solve systems of equations that they themselves have derived.

## Collection Contents

### Introducing Systems of Equations -- Solving by Accurate Graphing

by Jesse Mercer

This lesson introduces the concept of systems of linear equations by relying on the graphical interpretation. Students will come to understand that a system is "two or more equations plotted on the same set of axes". In subsequent lessons, this concept will be developed and nuanced to include any two equations whose variables represent the same quantities.
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### Solving Systems with the Substitution Method

by Jesse Mercer

This lesson addresses the substitution method of solving systems of equations. The students are introduced to the method through the use of a "real-world" problem.
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### Solving systems using linear combination

by Jesse Mercer

This lesson addresses the third method of solving systems of equations: linear combination. After learning the method and practicing it, students will be involved in a brief discussion of how to decide which method to use when confronted with a particular system of equations.
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### Applying systems of equations -- finding break-even points

by Jesse Mercer

In this lesson, students will apply the methods they have been learning to the data they have collected for their project. They will use the cost curve they generated earlier in the project and combine it with five different possible revenue curves (representing potential prices they can charge for their product). This will let them know how many units they will need to sell at various prices in order to make an informed decision about how much to charge for their product. This lesson will take at least 2, potentially 3 or 4, 45-minute class periods to complete.
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### Using TI-82/83+ Calculators to Solve Systems

by Jesse Mercer

This lesson exposes students to a new functionality of the TI-82/83+ calculator: solving systems of equations. Students practice solving systems using provided examples and then verifying their own data.
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### Preparing a loan presentation

by Jesse Mercer

In this lesson, students work collaboratively to prepare a loan proposal presentation for the school principal, using the data they have gathered for their project. This lesson will take several 45-minute class periods, depending upon how much outside work the students are assigned. It also requires some previous communication with a member of the administration to schedule their appearance in class.
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