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 Eilmination

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.

Graphing Systems of Equations

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?

All About Linear Graphing

Topics for Today:

No new material was covered in class today; instead, we reviewed finding slope, using the various forms of a line, and finding parallel and perpendicular lines in preparation for our test tomorrow over the chapter.  We did not cover the last 2 sections of the chapter and they will not be on the test.

To be proficient in the concepts covered in this chapter students will need to be able to move freely among the different line forms.

Students were provided a graphic organizer that includes all the main topics from this chapter up to this point.  We have one more topic on Absolute Value Translations to cover. 

Sections Covered in Textbook:

This is a summary of Chapter 6 so far....

Resources & Tutorials:

Graphing Absolute Value

Topics for Today:

Today we continued our discussion of graphing absolute value functions through a DESMOS activity in class.  The purpose of the activity was to allow students to explore the various parts of an absolute value equation and to draw their own conclusions about how changing various pieces of the equation affects how the graph looks.  All of these variations are called translations.

Vocabulary: absolute value graph, translation, vertex

Sections Covered in Textbook:

6-7: Graphing Absolute Value Equations (pages 325-329)

Resources & Tutorials:

1)  Graphing Absolute Value Equations Introduction
2)  Shifting Absolute Value Graphs
3) Desmos Activity - Exploring Absolute Value (see Google Classroom for code)

4) Absolute Value Translations Notes 

Parallel and Perpendicular Lines

Topics for Today:

The slope of two lines can produce a special relationship between those lines.  Two such relationship are parallel lines and perpendicular lines.  Parallel lines exist in the same plane but will never intersect, and they always have the same slope.  Perpendicular lines are special because when they intersect, the lines form 90° angles.  The slopes of perpendicular lines are negative reciprocals of each other, and when those slopes are multiplied together, the result is -1.

We will be analyzing the slopes of two lines to determine if either relationship exists, and we will be deducing linear equations from a given point that is either parallel or perpendicular to the given line.

Parallel and perpendicular lines are always determined by the relationship of their slopes!

Vocabulary: parallel lines, perpendicular lines, negative reciprocal

Sections Covered in Textbook:

6-5: Parallel and Perpendicular Lines (pages 311 - 316)

Resources & Tutorials:

1) How do you find the slope of a line if you have a parallel line?
2)  How do you write an equation of a line in slope-intercept form if you have one point and a parallel line?
3) How do you find the slope of a line if you have a perpendicular line?
4) How do you write an equation of a line in slope-intercept form if you have one point and a perpendicular line?
5) How to tell if lines are parallel, perpendicular, or neither.

Point-Slope Form

Topics for Today:

Today we discussed the last line form, point-slope form, through a PowerPoint presentation (link below).  

The three different forms of a linear equation are all useful for different reasons.  Each one has a pattern to follow.  Once the patterns are mastered, linear graphing becomes much easier.  

Vocabulary: point-slope form

Sections Covered in Textbook:

6-4: Point-Slope Form and Writing Linear Equations (pages 304-309)

Resources & Tutorials:

1) What is the point-slope form of a linear equation?

2) How do you write an equation of a line in point-slope form if you have the slope and one point?

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3) Exploring Point-Slope Form (PowerPoint)

Standard Form

Topics for Today:

Our discussion about linear equations continued today.  We have already explored slope and slope-intercept form of a line.  Today, we looked at a different form - standard form.  The standard form of a line is defined as a linear equation such that

Ax + By = C

A, B, and C must all be integers.

A must be positive.

Although it's easy to visualize a line that is in slope-intercept form (the form we worked with yesterday), it's very easy to find both the x- and y-intercepts when a line is in standard form.  These intercepts are where the line crosses the x- and y-axes, when one of our coordinates is zero.  Solving the equation when substituting a zero for a value is a quick process, because multiplying by zero removes the variable from the equation.  Once we find our intercepts, it's very easy to graph our equation.

Standard form can be nice for students who are not fond of working with fractions, and we'll be using standard form when we move to solving systems of equations in the next chapter.  In addition, many of the other graphs that students will see in later mathematics classes are written in standard form with the variables all on one side of the equation.  Comfort with standard form will help students cope when they are introduced to more complicated equations.

Vocabulary: standard form of a line, x-intercept

Sections Covered in Textbook:

6-3: Standard Form (pages 298-302)

Resources & Tutorials:

1) What is the standard form of a linear equation?
2) How do you use x- and y-intercepts to graph a line in standard form?

Slope-Intercept Form

Topics for Today:

One of the most recognizable forms of a line is the slope-intercept form.  This line form is very useful because it's easy to visualize the actual line simply by looking at the equation.  From slope-intercept form, you can tell if the slope is positive or negative, and if the slope is steep or shallow, and also it demonstrates where the line crosses the y-axis (this is the y-intercept).

We talked about what an intercept is (this word sounds an awful lot like intersect!).

I reminded students of our work on solving literal equations - this skill will be especially helpful for our unit on linear equations, as we'll be looking at three different forms for a linear equation.  To put a line in slope-intercept form, simply solve for the variable "y".

Slope intercept form looks like this:  y=mx + b

• m is the slope
• b is the y-intercept 

Vocabulary: linear equation, y-intercept, slope-intercept form

Sections Covered in Textbook:

6-2: Slope-Intercept Form (pages 291-296)

Resources & Tutorials:

1) What is a linear equation?
2) What is the y-intercept?
3) What is the slope-intercept form of a line?

Rate of Change and Slope

Topics for Today:

We will continue our discussion about functions as we explore linear functions (lines).  ALL LINES (with the exception of vertical lines) are functions.  This unit will cover many different aspects of line, beginning with rate of change, otherwise known as slope.  We associate slope with the "steepness" of a line.  Slopes can be positive, negative, zero, or undefined.

Slope is a 2-dimensional concept.  We will see how fast something rises (goes up) compared to how fast it travels in a horizontal direction.  Slope is defined as the change in the y-coordinate divided by the change in the x-coordinate.  To calculate slope, you need any two points on a line.  It does not matter where you start as long as you start in the same place for each component.

Vocabulary: rate of change, slope

Sections Covered in Textbook:

6-1: Rate of Change and Slope (pages 282-289)

Resources & Tutorials:

1) What does the slope of a line mean?
2) How do you find the slope of a line from two points?
3) How do you find the slope of a line from a graph?