Applications of Systems Part II

Topics for Today:

Yesterday we began to tackle applications of systems of equations.  Basically, we are going to be solving story problems that have two unknowns, requiring us to write two equations to solve them.  These types of problems can be categorized and patterns emerge as we see more and more of these types of problems.  

We did a Desmos activity today that helped us with building equations and solving them.  

Sections Covered in Textbook:

7-4: Applications of Linear Systems (pages 362-368)

Resources & Tutorials:

1) How do you solve a word problem using two equations? 
2) Simple word problem resulting in two equations (not a video)

3) Desmos Activity on Systems of Equations (Login with Google)

Applications of Systems

Topics for Today:

One of the things that is most annoying about Algebra I is the focus on the processes and procedures for solving equations, inequalities, and problems.  Most of the time we are focused on process rather than application, but this foundational toolset is critical to solving problems requiring higher thinking and reasoning.

Today we used our knowledge of solving systems of equations to solve some real-world problems.  Typically students lack confidence when solving story problems, although it is these very problems where we get to use all the skills we have been building.  As I continually reinforce to our students, mastering Algebra requires repetition and practice, like any other skill we hope to master.  The only way to become competent and confident solving story problems is to do them -- LOTS of them.

The main thing to remember when solving the linear systems we have been working on is that if we have two variables, we will need two equations to solve.  The same would be true for three variables (a topic for Algebra II where you need three equations).

For these story problems, first, we must identify and define our variables.   Second, we will analyze the given information and write our equations based upon the given information.  Once we have our equations, we can determine the best method to solve the system.  Finally, we must look at the question that was asked and make sure that our solution answers the question, that we have the correct units, and that our answer makes sense.

Many of these story problems follow a pattern, and identifying the pattern makes the problem easier.  For this topic, we normally have several patterns to choose from:  mixtures, distance-rate-time (these can come in many forms, and can deal with things like water and wind currents that speed up or slow down the traveler), and break-even.

Sections Covered in Textbook:

7-4: Applications of Linear Systems (pages 362-368)

Resources & Tutorials:

1) How do you solve a word problem using two equations? 
2) Simple word problem resulting in two equations (not a video)

Systems of Linear Inequalities

Topics for Today:

We expanded our discussions about linear inequalities and systems to include the topic of systems of linear inequalities.  We discovered in our lesson yesterday that linear inequalities include all the points on one side of a border.  When we combine two linear inequalities, we are going to look for where both overlap.  The only way to represent this overlap region is by graphing.  (Recall that we discussed and practiced three different ways of solving linear systems - graphing, substitution method, and elimination method.)

The solution to the system

y < 2x + 1 and

y > 1/2 x -3 

looks like this:

The red region represents the overlap,

and therefore the solution to the system.

Vocabulary:  system of linear inequalities, solution of a system of linear inequalities

Sections Covered in Textbook:

7-6: Systems of Linear Inequalities (pages 377-384)

Resources & Tutorials:

1) What is a system of inequalities? 
2) How do you solve a system of inequalities by graphing?

Linear Inequalities

Topics for Today:

We are still finding our way around the coordinate plane, and today we discussed how to graph the solution to a linear inequality.  A linear inequality describes a region of the coordinate plane that has a boundary line.  The solution to a linear inequality are all the coordinate points that make a linear inequality true.  The solutions to inequalities contain infinitely many more solutions than that of equations, and the same is true for linear inequalities.

Solutions to linear inequalities involve graphs.  The process for graphing linear inequalities is very similar to graphing linear equations with a few additional details.  The basic process for graphing linear inequalities is

1. Treat the inequality just like an equation.  Use the equation to graph the boundary line.
2. Determine if the boundary line is a part of the solution
• For equations that are strictly greater or less than (>  or <), the boundary is NOT included and should therefore be a dashed line.
• For equations that are greater than or equal to or less than or equal to (≥ or ≤) the boundary line IS included and should therefore be drawn as a solid line.  
3. Next, determine which side of the line the solution points fall.  The best way to accomplish this is to pick a point on either side, and test the inequality for truth.  The point that generates a true statement is on the side of the line with the solution.
4. Once the correct side of the boundary is found, shade this region to indicate where the solutions are.  

Vocabulary: linear inequality, solution of an inequality

Sections Covered in Textbook:

7-5: Linear Inequalities (pages 370-375)

Resources & Tutorials:

1. What is a linear inequality?
2. How do you figure out if the boundary is part of the graph of the inequality?

Solving Systems Using Elimination

Topics for Today:

We are still working on systems of linear equations.  Today, we discussed elimination method, and with a system of two equations, this method is really the preferred one.

Steps for Solving Using Elimination Method

1. In your original system, make sure both equations are in the same form (standard form works best!).  Line your equations up so the variables are aligned in columns.  
2. Determine which variable should be eliminated.  Look for matching numbers and opposite signs or create them using multiplication.  You may have to multiply both equations so that you can eliminate one variable.  
3. Eliminate the chosen variable.  Solve for the other variable.  
4. Take the value you found in Step 3 and substitute it into one of the original equations to solve for the other variable. 
5. Identify your solution – it will be an ordered pair!
6. Check both original equations with the solution you found.  

 Graphic Credit: http://www.hotelsrate.org/solving-systems-of-linear-equations-by-elimination-examples/

Vocabulary:  elimination method

Sections Covered in Textbook:

7-3: Solving Systems Using Elimination (pages 353-359)

Resources & Tutorials:

1. How do you solve a system of equations using the elimination by addition method? 
2. How do you solve a system of equations using the elimination by multiplication method?
3. What's another way of solving a system of equations using the elimination by multiplication method?

Solving Systems Using Substitution

Topics for Today:

We are still working on solving systems of equations and introduced a new method today that uses algebra instead of graphing.  We have discussed the limitations of the graphing method, and why we might use algebra instead.  There are three algebraic methods used to solve systems:  substitution method, elimination method, and the matrix method.  We will learn and practice the first two; solving of matrices with systems of 2 or more equations is covered in Algebra II.

The general process for solving systems algebraically is the same.  First, we solve for one of the variables; then we substitute that solution into one of our equations to find the second variable.

For the substitution method, we follow this process:

1. In the original system, see if one variable is isolated; if not, then isolate a variable. 
2. Substitute the expression into the second equation.
3. Solve the equation for the first variable.
4. Substitute the solution found in step 3 into one of the original equations to solve for the other variable. 
5. Identify the solution as an ordered pair.
6. Check both original equations to ensure the solution works for both. 

(Graphic Credit: https://www.mathwarehouse.com/algebra/linear_equation/systems-of-equation/solve-by-substitution.php)

Vocabulary:  substitution method

Sections Covered in Textbook:

7-2: Solving Systems Using Substitution (pages 347-351)

Resources & Tutorials:

1) How to solve a system using substitution method.
2) Solving Systems of Equations by Substitution.

Solving Systems Using Graphing

Topics for Today:

We are still working with graphing linear equations, but we've expanded our conversation to include systems of linear equations.  A linear system of equations is simply two or more linear equations containing the same variables.  When we deal in generic equations, we almost always use the variables x and y; however, when we use systems to solve real problems, we may define our variable with different letters that better match our problem.  For instance, if we are talking about costs and revenue, we may choose to use c and r for our variables.

Systems of two linear equations have three possible types of solutions because they are based upon where two lines intersect on a plane:  they either intersect nowhere, intersect at one point, or intersect at every point.  If there is a solution, it is represented as an ordered pair.

Summary of Systems of Equations
(Click Graphic to Enlarge)

Vocabulary:  system of linear equations, solution of a system of linear equations

Sections Covered in Textbook:

7-1: Solving Systems by Graphing (pages 340-345)

Resources & Tutorials:

1) What is a system of linear equations?
2) How do you solve a system using graphing?
3) What is a solution to a system of equations? 
4) What are the three types of solutions to a system of equations?