Lesson 1 Row by Row Solidify Understanding
Learning Focus
Extend strategies for solving systems in two variables to solve systems in
What is the most efficient way to solve systems of equations that involve
Open Up the Math: Launch, Explore, Discuss
Carlos likes to buy supplies for the twins’ business, Curbside Rivalry, at the All a Dollar Paint Store where the price of every item is a multiple of
Carlos and Clarita are trying to figure out the cost of a gallon of paint, the cost of a paintbrush, and the cost of a roll of masking tape based on the following purchases:
Week 1: Carlos bought
Week 2: Carlos bought
Week 3: Carlos bought
1.
Determine the cost of each item using whatever strategy you want. Show the details of your work so that someone else can follow your strategy.
You probably recognized that this problem could be represented as a system of equations. In previous math courses you have developed several methods for solving systems: graphing, substitution, elimination, and row reduction of matrices.
2.
Which of the methods you have developed previously for solving systems of equations could be applied to this system? Which methods seem more problematic? Why?
Previously, you learned how to solve systems of equations involving two equations and two unknown quantities using row reduction of matrices.
3.
Modify the “row reduction of matrices” strategy so you can use it to solve Carlos’ and Clarita’s system of three equations using row reduction. What modifications did you have to make, and why?
Pause and Reflect
4.
Decide on a reasonable context and write a story for the information given in the following matrix:
5.
Solve for the unknowns in your story by using row reduction on the given matrix. Check your results in the story context to make sure they are correct.
Ready for More?
Compare the steps for solving a system by using row-reduction of matrices versus using substitution by rewriting the matrix from problem 4 as a system of equations, and then solving the system by substitution.
Written as a system of linear equations:
Solve using substitution as the solution method:
Takeaways
The standard procedure for row-reducing an augmented matrix can be explained by my answers to these questions:
Why is getting
Why do I want to get
Why might switching rows of the matrix be helpful?
Adding Notation, Vocabulary, and Conventions
The procedures we use for solving systems of linear equations by elimination can be replicated by the row reduction steps for matrices. These include:
, represented as
. , represented as
. , represented as
.
Vocabulary
- augmented matrix
- identities (additive/multiplicative) with matrices
- matrix row reduction
- row reductions of matrices
- Bold terms are new in this lesson.
Lesson Summary
In this lesson, we reviewed strategies for solving systems of equations and extended them to solve systems with more that
1.
Solve the system of equations using an algebraic method.
2.
How do you determine when a system of equations has no solution?