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K12MATH011: Algebra II

Unit 2: Linear Systems   In unit 2, you will move from simple linear equations and inequalities to begin an exploration of more complex systems of equations and inequalities. You will discover various methods used to solve these equations and inequalities, in two variables. You will also learn about linear programming. Unit 2 ends with an introduction to systems of three equations, demonstrating that the techniques already learned may be applied to systems of higher and higher dimension. Once you understand the methods demonstrated, you will learn about matrices and methods of solving systems using matrix algebra. Matrices are rows and columns of numbers. For us, the numbers are the coefficients and constants from linear equations. The methods used to solve these systems are extremely helpful in later courses. You will complete the unit with an introduction to vectors, which are rows or columns from matrices. Learning how to use vectors is very useful in business and science.
 
Solving linear systems involves finding common points for two or more lines. These are shared solutions among the equations, and the techniques learned have important roles to play in solving problems in business and science. Solving linear inequalities involves finding all common points among the relations.

Unit 2 Time Advisory
Completing this unit should take approximately 22 hours and 15 minutes.

☐    Subunit 2.1: 1 hour and 15 minutes

☐    Subunit 2.2: 5 hours and 45 minutes

☐    Subunit 2.3: 2 hours and 30 minutes

☐    Subunit 2.4: 2 hours and 30 minutes

☐    Subunit 2.5: 30 minutes

☐    Subunit 2.6: 6 hours

☐    Subunit 2.7: 3 hours and 45 minutes

Unit2 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Graph systems of two equations in two variables to find a solution to the system. - Use algebraic methods to solve systems of two equations. - Use algebraic methods to solve systems of three equations. - Graph and solve systems of linear inequalities. - Use linear programming methods to solve systems of equations. - Use matrix operations to simplify and solve systems of equations. - Perform transformations on matrices. - Find the determinant of a square matrix. - Use the determinant of a square matrix to solve a system of equations via Cramer’s rule. - Identify a given vector’s graph.

Standards Addressed (Common Core): - CCSS.Math.Content.HSA-REI.C.5 - CCSS.Math.Content.HSA-REI.C.6 - CCSS.Math.Content.HSA-REI.D.11 - CCSS.Math.Content.HSA-REI.D.12 - CCSS.Math.Content.HSF-IF.B.5 - CCSS.Math.Content.HSN-VM.C.6 - CCSS.Math.Content.HSN-VM.C.7 - CCSS.Math.Content.HSN-VM.C.8 - CCSS.Math.Content.HSN-VM.C.9 - CCSS.Math.Content.HSN-VM.C.12 - CCSS.ELA-Literacy.RST.9-10.5 - CCSS.ELA-Literacy.RST.9-10.9 - CCSS.ELA-Literacy.RST.11-12.5 - CCSS.ELA-Litearcy.RST.11-12.9 - CCSS.ELA-Literacy.WHST.11-12.1a - CCSS.ELA-Literacy.WHST.11-12.2e

2.1 Solving Linear Systems by Graphing   There are three possible solutions to a linear system: no point (i.e., no solution), single point, or infinite points. Either two lines meet at one point, overlap (i.e., represent the same line), or don’t meet (are parallel). We begin studying the solutions of systems by first looking at graphs, since solutions are easier to see this way.

  • Explanation: Khan Academy’s “Solving Systems Graphically” Link: Khan Academy’s “Solving Systems Graphically” (YouTube)
     
    Instructions: This video discusses the process of solving systems of equations by graphing. Solving is a method for finding common solutions for two or more equations. Graphing is a great way to begin solving linear systems, since it is a visual approach to such solutions, making the concept easy to see. However, when the common solution is not made up of integer coordinates, it becomes harder to find graphically.
     
    Watching the video and writing notes should take approximately 15 minutes.
     
    Standards Addressed (Common Core):

     
    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy.

  • Explanation: Khan Academy’s “Example 2: Graphically Solving Systems” Link: Khan Academy’s “Example 2: Graphically Solving Systems” (YouTube)
     
    Instructions: This video continues discussing the process of solving systems of equations by graphing. In this example, we see that not all systems have a solution. This is easier to see graphically than by algebra, and that is why we approach the topic with graphing first.
     
    Watching the video and writing notes should take approximately 15 minutes.
     
    Standards Addressed (Common Core):

     
    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United States License. It is attributed to Khan Academy. 

  • Web Media: GeoGebra: “System of Linear Equations” Link: GeoGebra: “System of Linear Equations” (HTML)
     
    Instructions: This interactive graph allows you to change several factors to affect the graph of a system of two linear equations. You may use a slider system for both equations to change slope and y-intercepts. Note how the lines and the common solution are both affected.
     
    Practicing with this application and writing notes should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

     
    Terms of Use: This resource is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License. 

2.2 Solving Linear Systems Algebraically   As with solving equations, graphs are sometimes not as useful for systems, as the common solution may not have integer coordinates. In this case, algebraic methods are not only more useful, but they also can be translated to programming for computer use. Unlike graphs, you may not be able to see which of the three solution types a system may have. With practice, however, you may be able to tell just by looking at the equations whether they have a solution, and if they do, whether the solution is a single point or an entire line.

2.3 Systems of Linear Inequalities   Linear inequalities are special cases of relations that have a lot of application in the real world. Manufacturing, for instance, has many examples of processes that may be modeled by systems of inequalities. Engineering uses some systems as well. Learning how to solve these relations is very useful.

2.4 Linear Programming   Linear programming is a special application of solving systems of linear inequalities. In linear programming, you will learn that certain aspects or processes have measurable constraints, and we can use those constraints to find limits in a system (that is, the extent of our resources limits how we may use them, and these limits form boundary lines). We can then use corner points of the graphs of the system, using these limits, to solve maximum/minimum problems in resource distribution, for instance. That is, we can answer the following question: What is the most effective use of our resources when we have more than one way to use them? A good example is whether to use raw materials to build cars or minivans. Linear programming can tell us the number of each to produce to maximize revenue, based on the constraints of the system.

2.5 Systems of Equations in Three Variables   In real life, few problems are simple enough to be expressed with only two variables, but the techniques we use for larger systems are just extensions of what we have already learned. Here we look at systems in three variables and see that solving larger systems is just a matter of using more steps.

2.6 Introduction to Matrices   Now that we have looked at systems and have begun solving them, let us examine them again in terms of matrices. A matrix is a mathematical structure that has centuries of development behind it. Long before anyone had a computer, people had to have the means to solve large systems. As such, some of the greatest minds in mathematics sought to find new ways to simplify these systems. Lots of theory on matrices has led to quicker methods for solving systems based on them and has provided us with methods that have proven easy to write in terms of computer programming today.

2.6.1 Definition of a Matrix and Matrix Operations   - Explanation: Khan Academy’s “Introduction to the Matrix” Link: Khan Academy’s “Introduction to the Matrix” (YouTube)
 
Instructions: This video contains an excellent introduction to matrices, including notations and basic operations.
 
Watching the video and writing notes should take approximately 15 minutes.
 
Standards Addressed (Common Core):

-   [CCSS.Math.Content.HSN-VM.C.7](http://www.corestandards.org/Math/Content/HSN/VM/C/7)
-   [CCSS.Math.Content.HSN-VM.C.8](http://www.corestandards.org/Math/Content/HSN/VM/C/8)

   
 Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 United States
License](http://creativecommons.org/licenses/by-nc-sa/3.0/us/). It
is attributed to Khan Academy. 

2.6.2 Matrix Transformations   - Explanation: Dr. Carol J. V. F. Burns’s One Mathematical Cat, Please! Topics in Algebra II: “Graphing Tools: Vertical and Horizontal Translations” Link: Dr. Carol J. V. F. Burns’s One Mathematical Cat, Please! Topics in Algebra II: “Graphing Tools: Vertical and Horizontal Translations” (HTML)
 
Instructions: This page contains good, succinct information on vertical and horizontal shifts of geometric objects using matrix operations. This is information used in many fields, including physics and engineering, manufacturing, and design and business decision making. Quite often, objects and processes may be created by transforming simpler objects. Simpler objects transformed this way keep many of the same properties, such as if a point on a simple object is to the left of another, it will remain to the left of the other even when the object is moved or stretched. This knowledge allows us to study the simpler forms first. Note: Practice problems are on the bottom of the page for self-testing.
 
Completing the activity and writing notes should take approximately 45 minutes.
 
Standards Addressed (Common Core):

-   [CCSS.Math.Content.HSN-VM.C.8](http://www.corestandards.org/Math/Content/HSN/VM/C/8)
-   [CCSS.Math.Content.HSN-VM.C.9](http://www.corestandards.org/Math/Content/HSN/VM/C/9)

   
 Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial 2.5
License](http://creativecommons.org/licenses/by-nc/2.5/). It is
attributed to Dr. Carol JVF Burns.

  • Explanation: Dr. Carol J. V. F. Burns’s One Mathematical Cat, Please! Topics in Algebra II: “Graphing Tools: Vertical and Horizontal Scaling” Link: Dr. Carol J. V. F. Burns’s One Mathematical Cat, Please! Topics in Algebra II: “Graphing Tools: Vertical and Horizontal Scaling” (HTML)
     
    Instructions: This page contains good, succinct information on vertical and horizontal stretching and compressing of geometric objects using matrix operations. Note: Practice problems are on the bottom of the page for self-testing.
     
    Completing the activity and writing notes should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

     
    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. It is attributed to Dr. Carol JVF Burns.

  • Explanation: Dr. Carol J. V. F. Burns’s One Mathematical Cat, Please! Topics in Algebra II: “Graphing Tools: Reflections and the Absolute Value Transformation” Link: Dr. Carol J. V. F. Burns’s One Mathematical Cat, Please! Topics in Algebra II: “Graphing Tools: Reflections and the Absolute Value Transformation” (HTML)
     
    Instructions: This page contains good, succinct information on reflections and absolute value transformations of geometric objects using matrix operations. Note: Practice problems are on the bottom of the page for self-testing.
     
    Completing the activity and writing notes should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

     
    Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. It is attributed to Dr. Carol JVF Burns.

  • Web Media: GeoGebra: “Matrix Transformation” Link: GeoGebra: “Matrix Transformation” (HTML)
     
    Instructions: This page allows you to practice using matrices to change size and scale and to reflect an object across an axis. The interactive graphic is handy for demonstrating how matrix operations can be used to change the shape of geometric forms, which can be useful in simulations, video game creation, and graphic design.
     
    Practicing with this application and writing notes should take approximately 45 minutes.
     
    Standards Addressed (Common Core):

     
    Terms of Use: This resource is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

2.6.3 The Determinant of a Square Matrix and Cramer’s Rule   - Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 8, Section 5: Determinants and Cramer’s Rule” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 8, Section 5: Determinants and Cramer’s Rule” (PDF)
 
Instructions: This precalculus text has a detailed, though somewhat technical, explanation for using Cramer’s rule. Read pages 614–622, and then try the problems at the end of the chapter. Answers are on the pages that follow.
 
Completing the activity and writing notes should take approximately 45 minutes.
 
Standards Addressed (Common Core):

-   [CCSS.Math.Content.HSN-VM.C.12](http://www.corestandards.org/Math/Content/HSN/VM/C/12)

   
 Terms of Use: This resource is licensed under a [Creative Commons
Attribution-NonCommercial-ShareAlike 3.0 Unported
License](http://creativecommons.org/licenses/by-nc-sa/3.0/us/http:/creativecommons.org/licenses/by-nc-sa/3.0/deed.en_US).
It is attributed to Carl Stitz and Jeff Zeager.

2.7 An Introduction to Vectors   Vectors are a way to represent the rows or columns of a matrix. As with matrices, a great deal of thought has gone into operations with vectors, and as with matrices, we now have ways to rewrite problems using vectors that computers can solve.