Lesson 1 Row by Row Solidify Understanding

Learning Focus

Extend strategies for solving systems in two variables to solve systems in $n$ variables.

What is the most efficient way to solve systems of equations that involve $n$ variables and $n$ linear equations?

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 $$ 1$ . This makes it easy to keep track of the total cost of his purchases. Clarita is worried that items at All a Dollar Paint Store might cost more, so she is going over the records to see how much Carlos is paying for different supplies. Unfortunately, Carlos has once again forgotten to write down the cost of each item he purchased. Instead, he has only recorded what he purchased and the total cost of all of the items.

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 $2$ gallons of paint and $1$ roll of masking tape for $$ 30$ .

Week 2: Carlos bought $1$ gallon of paint and $4$ brushes for $$ 20$ .

Week 3: Carlos bought $2$ brushes and $1$ roll of masking tape for $$ 10$ .

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:

$[\begin{matrix} \begin{array}{r} 1 & 2 & 2 \\ 2 & 1 & 3 \\ 1 & 2 & 1 \end{array} & | \begin{array}{r} 16 \\ 17 \\ 13 \end{array} \end{matrix}]$

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.

$[\begin{matrix} \begin{matrix} 1 & 2 & \begin{matrix} 2 & 16 \end{matrix} \end{matrix} \\ \begin{matrix} 2 & 1 & \begin{matrix} 3 & 17 \end{matrix} \end{matrix} \\ \begin{matrix} 1 & 2 & \begin{matrix} 1 & 13 \end{matrix} \end{matrix} \end{matrix}]$

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 $1$ ’s on the diagonal helpful?

Why do I want to get $0$ ’s in the non-diagonal positions of one column before I move on to working with getting a $1$ in the next column?

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 $k R_{1} \to R_{1}$ .
, represented as $k R_{1} + R_{2} \to R_{2}$ .
, represented as $R_{2} \leftrightarrow R_{1}$ .

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 $2$ variables. We focused specifically on the process of using matrices and row-reduction to solve systems.

Retrieval

1.

Solve the system of equations using an algebraic method.

${\begin{cases} - 4 x + y = 3 \\ x + y = - 2 \end{cases}$

2.

How do you determine when a system of equations has no solution?