# UTPA STEM/CBI Courses/Calculus/Applied Linear Algebra

Course Title: Linear Algebra

Lecture Topic: Linear Equations

Instructor: Cristina Villalobos

Institution: University of Texas-Pan American

## Backwards Design

Course Objectives

• Primary Objectives- By the next class period students will be able to:
• Model application problems as a set of linear equations
• Determine if the linear system of equations has none, one, or multiple solutions

• Sub Objectives- The objectives will require that students be able to:
• Identify the variables and determine the coefficients in the system of linear equations
• Determine the domains for the variables
• Identify when systems have none, one, or infinite number of solutions
• Difficulties- Students may have difficulty:
• Identifying variables
• Identifying coefficients in a linear system
• Real-World Contexts- There are many ways that students can use this material in the real-world, such as:
• Modeling the constraints (linear equations) in a company
• Modeling chemical reactions

Model of Knowledge

• Concept Map
• Variables, Coefficients
• Linear equations
• Solutions
• Content Priorities
• Enduring Understanding
• Solving linear equations and determining if the system has none, one, or infinite number of solutions
• Modeling problems using linear equations
• Important to Do and Know
• Understand the theorem that states every linear system has none, one, or infinite number of solutions
• Identify systems of linear equations versus systems of nonlinear equations.
• Worth Being Familiar with
• Understanding the geometry behind the solution set of a system of linear equations
• Understanding what it means to solve a system of linear equations
• Checking if a set of values solves the linear equations

Assessment of Learning

• Formative Assessment
• In Class (groups)
• Given a word problem, students will identify the variables and coefficients in modeling a problem as a system of linear equations
• Students will then be asked to determine the solution set of a system of linear equations
• Homework (individual)
• Students will be asked to model a problem as system of linear equations
• Using eWiley students will be assigned more online homework problems
• Summative Assessment
• Students will be given an exam to assess their skills in solving linear systems and in appropriately determining the solution set

## Legacy Cycle

OBJECTIVE

By the next class period, students will be able to:

• Model application problems as a set of linear equations
• Determine if the linear system of equations has none, one, or multiple solutions

The objectives will require that students be able to:

• Identify the variables and determine the coefficients in the system of linear equations
• Determine the domains for the variables
• Identify when systems have none, one, or infinite number of solutions

THE CHALLENGE

You work for a start-up company that is housed in three floors of the Chrysler Building in New York City. Your task is to distribute efficiently a limited amount of resources to the three floors given the cost of each resource and the total amount of funds allocated to each floor.

GENERATE IDEAS

Students generate ideas on the type of resources to allocate and the amount provided to each floor given a certainly availability of funds.

MULTIPLE PERSPECTIVES

Students will be asked to obtain a video of a CEO discussing some constraints they consider when allocating resources. Students will also be asked to model their allocations and resources as a system of linear equations.

RESEARCH & REVISE

Students will conduct research to determine how to model their problem. The students need to determine what the variables and coefficients represent in their model. Then the students need to solve their model. In doing so, they will learn that systems of linear equations have none, one, or infinite number of solutions. Then they will need to focus on the system of linear equations that has a solution—after all the students were hired to allocate resources to each floor.

Students will be given homework problems that illustrate the various possibilities for solutions (none, one, or infinitely many).

GO PUBLIC

The students will present their model and describe the variables, the coefficients, and the linear equations. Moreover, they will interpret the solution. The students will be tested on solving linear system of equations through both open-ended and multiple-choice questions. Sometimes students know how to solve linear systems of equations but they may forgot how to determine if some values are a solution to the system of equations.

## Pre-Lesson Quiz

1. Is the point (3,4) a solution to the system of linear equations?

```  2x + y = 7
-3x + 2y = 12
```

2. Interpret a given system of linear equations and discuss what the system models.

3. Solve a system of linear equations provided by the instructor.

1. Is the point (-2,5) a solution to the system of linear equations?

```   4x + 2y = 2
5x  - y = -15
```

2. Is the point (1/2, 8) a solution to the system of linear equations?

```                -12x + 4y = 24
4x  - 2y = 6
```

3. Solve the following system of linear equations.

4. With the given problem, model it as a 3x3 system of linear equations.

5. Identify the coefficient matrix and the variables for the following system of linear equations.

```          4x + 9y – z = 8
2x + 3y + 7z = 28
-x + 8y – 14z = 0
```