 # 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.

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.

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.

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.

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.

• 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.

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.

• Explanation: Everything Maths: “Solving Simultaneous Equations” Link: Everything Maths: “Solving Simultaneous Equations” (YouTube and HTML)

Instructions: This video explains simultaneous equations (systems) and provides written step-by-step instructions on how to perform the most common procedures for solving them. Included are substitution, elimination, and graphing.

Watching the video and writing notes should take approximately 30 minutes.

Instructions: This video contains an example of solving systems by elimination. This is an important method that can be extended to more than two equations. Pay close attention to the methods, as they serve as a good introduction to matrix methods that we will see later.

Watching the video and writing notes should take approximately 15 minutes.

Instructions: This video contains a more complex example of solving systems by elimination. This is an important method that can be extended to more than two equations. Note that the method involves applying equation properties in new ways.

Watching the video and writing notes should take approximately 15 minutes.

Instructions: This video contains a discussion of infinite common solutions. Thus, we see that systems may have only one common point, but they may also have all points in common.

Watching the video and writing notes should take approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Systems of Equations with Elimination” Link: Khan Academy’s “Systems of Equations with Elimination” (HTML)

Instructions: This is a self-quiz on adding and subtracting polynomials. Note that there are hints available if needed.

Taking the quiz and writing notes should take approximately 1 hour.

Instructions: This video discusses a method for solving systems of equations via substitution. This is another algebraic approach to solving systems and one we will see again in other units. This particular example shows that two parallel lines have no common solution. Now we have seen the third type of system. We have seen the one common point system (two lines intersecting at one point), the infinite common points system (two lines that are actually the same), and now the no common points system (parallel lines). Write a four-paragraph (or more) essay comparing and contrasting the substitution and elimination methods. What are the weaknesses and strengths of each method? Are there times when one has advantage over the other in terms of time and steps spent?

Completing this activity should take approximately 1 hour and 30 minutes.

Instructions: This video continues the approach of solving systems via substitution. This example shows a simple problem that is fairly easy to solve.

Watching the video and writing notes should take approximately 15 minutes.

Instructions: This video continues the approach of solving systems via substitution. This example shows a simple problem that is fairly easy to solve and then shows how to graph the system.

Watching the video and writing notes should take approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Systems of Equations with Substitution” Link: Khan Academy’s “Systems of Equations with Substitution” (HTML)

Instructions: This is a self-quiz on adding and subtracting polynomials. Note that there are hints available if needed.

Taking the quiz and writing notes should take approximately 45 minutes.

• Web Media: GeoGebra: “Solving Systems of Equations by Substitution” Link: GeoGebra: “Solving Systems of Equations by Substitution” (HTML)

Instructions: This is a self-check quiz. You are able to check solutions and try new systems.

Taking the quiz and writing notes should take approximately 45 minutes.

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.

Instructions: This video shows the best method for solving systems of linear inequalities, which is by graphing. Note that the instructor in the lesson uses both the comparison and the coordinate test method for each inequality. Either can be used; both need not be.

Watching the video and writing notes should take approximately 15 minutes.

Instructions: This video demonstrates graphing a solution for multiple constraints. As we will see, this does not make the problem more difficult. Comparison or coordinate tests can be used in this case, as well. This particular case shows an example with no common solutions.

Watching the video and writing notes should take approximately 15 minutes.

Instructions: This video demonstrates a system of inequalities from a word problem. Note carefully how the instructor constructs the inequalities.

Watching the video and writing notes should take approximately 30 minutes.

• Web Media: GeoGebra: “Enter a System of Two Linear Inequalities” Link: GeoGebra: “Enter a System of Two Linear Inequalities” (HTML)

Instructions: This interactive graph can be modified by changing the inequalities or entering new ones. This is great practice for seeing the visual link between the systems and their graphs. Note: The light green shading is for one inequality, the purple for the other, and the darker, greenish-blue is the common solution.

Practicing with this application and writing notes should take approximately 45 minutes.

• Web Media: GeoGebra: “Linear Equations and Inequalities – Systems” Link: GeoGebra: “Linear Equations and Inequalities – Systems” (HTML)

Instructions: This highly interactive graph can be modified by changing a variety of values and signs. There are many different inputs and possibilities to consider, but don’t let that discourage you. It’s a straightforward application. This is great practice not only for systems of inequalities and equations but also for linear equations. Also, note that there are three points that may be dragged to different areas to test for solutions.

Practicing with this application and writing notes should take approximately 45 minutes.

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.

• Explanation: YouTube: METAL Project’s Linear Programming: “Section 1: Introduction to Linear Programming” Link: YouTube: METAL Project’s Linear Programming: “Section 1, Introduction to Linear Programming” (YouTube)

Instructions: This video introduces you to linear programming, which is a technique often used to maximize or minimize some value. For instance, in this example, the instructor shows how two businesses maximize profits. The method begins with techniques we have seen for graphing linear inequalities, but instead of considering the shaded areas of a partial plane, linear programming looks at the vertices formed by the polygon created when the boundary lines are graphed.

Watching the video and writing notes should take approximately 30 minutes.

• Explanation: YouTube: METAL Project’s Linear Programming: “Section 2: Linear Programming – Final Two Steps” Link: YouTube: METAL Project’s Linear Programming: “Linear Programming – Final Two Steps” (YouTube)

Instructions: This video concludes the example given in the previous video. The final two steps are using the vertices and checking the final answer to determine whether it makes sense (i.e., the recommended checking of the final answer). This is a method commonly used in business and industry for maximizing output or profits and/or minimizing waste or costs.

Watching the video and writing notes should take approximately 30 minutes.

• Web Media: GeoGebra: “Linear Programming Example: Picturing Pictures” Link: GeoGebra: “Linear Programming Example: Picturing Pictures” (HTML)

Instructions: This linear programming example shows, graphically, how linear programming works. It is neither extensive, complicated to use, nor hard to understand, but after the videos this may be a good, light way to view feasibility and maximization. The program does all the work for you, but it would be good practice using the skills you’ve already learned to find the equations of the restraints and prove that your result is truly the optimal point of production.

Practicing with this application and writing notes should take approximately 45 minutes.

• Web Media: GeoGebra: “Linear Programming: Problem 1, Car vs. Moped” Link: GeoGebra: “Linear Programming: Problem 1, Car vs. Moped” (HTML)

Instructions: This relatively simple interactive graph allows you to move a point within the region of feasibility to find an optimal solution. Again, this doesn’t require any computation skills on your part, and it is good practice as it continues to reinforce the idea that the optimal point is always at a vertex on the region of feasibility. Note: The idea of this graph is that we can use some gas in the car and some in the moped, but which scenario will give us the best total mileage if both are used?

Practicing with this application and writing notes should take approximately 45 minutes.

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.

Instructions: This video introduces you to systems of three variables. Techniques for solving are extensions of techniques you have already learned, with additional steps. Real-world problems involve many variables that are solved with similar methods, but once you master the basics, you can solve any problem of this type.

Watching the video and writing notes should take approximately 15 minutes.

Instructions: This video contains another example of a system with three variables, with coefficients larger than one. It requires a few more steps of elimination to solve, but the technique only requires patience.

Watching the video and writing notes should take approximately 15 minutes.

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.

``````-   [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)

``````

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.

``````-   [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)

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.

• 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.

• 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.

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.

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

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.

Instructions: This is a video on the basics of vectors with a graphical representation. Different operations on vectors are covered, including scalar multiplications (which change the length of a vector), adding, and multiplying.

Watching the video and writing notes should take approximately 30 minutes.

Instructions: This is set of practice problems on scaling vectors. The set has a virtual scratch pad and a video reference on the right bottom (see “Stuck? Watch a video.”), for refreshing your understanding.

Completing the practice problems, watching the video, and writing notes should take approximately 1 hour.

Instructions: This is set of practice problems on adding vectors. The set has a virtual scratch pad and a video reference on the right bottom (see “Stuck? Watch a video.”), for refreshing your understanding.

Completing the practice problems, watching the video, and writing notes should take approximately 1 hour.

Instructions: This is a clever, yet simple, interactive graphic that demonstrates visually how vector addition works. By moving the endpoints around for vectors u and v, you can see the resulting vector change. It also illustrates why adding vectors is sometimes referred to as thetriangle method or the parallelogram rule. It also shows that if you want a force greater than you can produce straight on, you may be able to add smaller forces to achieve the same result.

Practicing with this application and writing notes should take approximately 45 minutes.