Solving Systems of Equations by 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?

5) Solving Systems by Graphing Class Notes

Assignments:

1) Solving Systems by Graphing Worksheet
2) Part 2:  Graphing Systems using DESMOS (Class Code RJ8WKK)

Topics for Today:

Today we finished our review of Chapter 6.  

Sections Covered in Textbook:

Chapter 6 (Pages 281-327)

* We did not do Section 6-6 - we will revisit this topic later on.

Resources & Tutorials:

See Blog entries for December 16th through January 25th

* Chapter 6 Review Sheet

* Review Sheet Answers

Assignments:

1) Chapter 6 Test tomorrow (7th) or Thursday (8th)

  *****Notebooks due

2) Finish Delta Math Weekly Review by end of Study Hall on Friday

Graphing Absolute Value Equations (7th Grade Only)

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

4) Absolute Value Translations Notes 

Assignments:

1) Finish Desmos Activity if not done in class.

2) Delta Math Assignment - Honors Alg 1 - Absolute Value Functions 

3) Delta Math Weekly Review: Jan 25 - 29 Algebra 1 Review due by Friday at 3pm 

4) Chapter 6 Test Wednesday (NOTEBOOKS DUE).

All About Linear Equations

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.

Sections Covered in Textbook:

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

Resources & Tutorials:

See Blog Entries from December 16th through January 20th for Videos & Resources

Assignments:

1) Delta Math Assignment: Honors Alg 1 - All About Linear Equations Practice 

2) Delta Math Jan 19-22 Algebra 1 Review due Friday at 3:15 pm

3) Absolute Value Graphing Activity due Monday (7th) - OR -- Exploring Linear Graphing Activity for 8th Grade due Monday.

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.

Assignments:

1) Delta Math Assignment: Honors Alg 1 - Parallel & Perpendicular Lines 

2) Delta Math Jan 19-22 Algebra 1 Review due Friday at 3:15 pm

Point-Slope Form

Topics for Today:

It was evident from the collected homework covering point-slope form that the students need more more work on this topic, and to approach the concept from another angle.

Today we revisited 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?

-----

3) Exploring Point-Slope Form (PowerPoint)

4) Point-Slope Form Class Notes

5) Watch a recording of today's class (see Google Classroom or HW email for password)

Assignments:

1) Point-Slope Form Worksheet

2) Chapter 5 Test Corrections due Thursday (7th) or Friday (8th)

3) Quiz Thursday (7th) or Friday (8th) over slope, slope-intercept form, and standard form.

4) Delta Math due Friday by 3pm.

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?

3) Standard Form Class notes

Assignments:

1) Standard Form worksheet

2) Quiz Thursday over Slope, Slope-Intercept Form, & Standard Form

3) Delta Math Review due Friday at 3 PM.

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?

4) Slope-Intercept Form Class Notes

Assignments:

1) Slope-Intercept Form Worksheet
2) Chapter 5 Test Corrections due Thursday

3) Delta Math Review Due Friday at 3 PM