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# Solving Systems of Equations using Inverse Matrix Operations

Lesson 7 of 12

## Objective: SWBAT model a system of equations using matrices and solve for the unknowns using inverse matrices.

## Big Idea: Students find their love for matrices when they learn how much easier solving systems of equations can be!

*45 minutes*

#### Exploration

*2 min*

To begin class, I ask students to answer the warm-up Clicker Questions** **located on page 2 of the Flipchart. Here, I ask students to EXPLAIN why we can do this. I will remind my students to explain using precise mathematical language. I often receive answers like, "Yes. It will solve for x. Then we get x=b/a.” In this instance, I will push students to use terms like "multiplicative inverse”, to name properties, etc.

Once students text in their answers, we will discuss the results as a class. If necessary, I will emphasize the fact that *a* and *1/a* are inverses, so when multiplied together they produce a factor of 1. At this point, Grade 12 students should be able to articulate this clearly.

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#### Explanation

*40 min*

Throughout this sections students will persevere is solving problems and will use matrices to reason abstractly and quantitatively (**MP1,*** ***MP2**). I plan to guide students through the notes on pages 4-12 of the Solving Systems using inverse matrices Flipchart. On page 5, I will ask students to practice converting each system to matrix form (**HSA-REI.C.8**). After giving them time to work, we will review the answers as a class and take the time to clarify any misconceptions.

Next, I will walk students through the example on page 6** (HSA-REI.C9).**** **If students are having difficulties accurately representing a system using matrices, I encourage them to complete each equation by using coefficients for all variables, including representing terms with coefficients of "0" and "1". For example, the system on Page 9 would look like this:

1x + 1y + 1z = 2

2x + 1y + 0z= 5

1x + 3y - 3z = 14

The examples on Page 7 provide the opportunity for students to practice. If time allows, I may do a* *Commit and Toss activity with these problems.

After giving the students the chance to demonstrate that they can set up the systems correctly using matrices, I will show my students how to solve linear systems using the matrix capabilities of their Ti graphing calculators *(***MP5**). I like to demonstrate this process using a Document Camera, by placing my calculator under it.

Page 10 introduces a real world application, the Bead Store Problem. I give students time to work on this problem and then we work together to verify the correct solution.

**Technology Tip for Using the Flipchart File:** Do not save the flipchart after clicking the yellow rectangles to make them disappear! Then, you can click the refresh page button at the end of every class period to reset the Flipchart back to its last saved position.

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Basic Matrix Operations
- LESSON 2: Matrices in the World Around Us
- LESSON 3: Determinants and Area of a Triangle (Day 1 of 2)
- LESSON 4: Determinants and Area of a Triangle (Day 2 of 2)
- LESSON 5: Inverses of Matrices
- LESSON 6: Applications of Inverses - Decoding
- LESSON 7: Solving Systems of Equations using Inverse Matrix Operations
- LESSON 8: Cramers Rule - Solving Systems of Equations
- LESSON 9: Determining the Matrix for Transformations
- LESSON 10: Matrix Transformations
- LESSON 11: Matrices Test Review
- LESSON 12: Matrices Exam