Math 10C

Chapter 8 Solving systems of linear equations graphically

Lesson 83 Number of Solutions for Systems of Linear Equations

Focus

The FOCUS section introduces the main concept that will be covered in the lesson.

TEXT Page 446

Read/Work through the  Focus  section at the top of page 446 in  your textbook.

Investigate

Investigate an idea or problem, with a partner if possible.
The investigation provides the background for the ideas presented in the lesson.

The investigation in the textbook is a good one, but will be pretty much impossible to do for students who are learning on their own.

I’ve created an alternative investigation (see below) that is essentially the same thing,  that you can do with just your computer.

TEXT Pages 446 - 447  (see below for an alternative)

Read/Work through the  Investigate  section on Pages 446 - 447 in your textbook.

A list of materials  you need for the Investigation is given in the side bar.

Your work will be a lot more efficient if you PLAN for and GATHER these items BEFORE you start the Investigation!

Watch

No videos were made for this.

Investigation

Introduction

At a secret location in northern Alberta,  far underground in a protected bunker,  mathematicians work with specially trained points to conduct experiments involving uniform motion.

In this scenario, two points, code-named “Red” and “Blue”,  move along a number line.

Based on initial conditions, the mathematicians attempt to predict the outcome of the race.

The simulation is shown below.
(take the scale to be in centimeters, and the time to be in seconds)

Check it out, then answer the questions that follow.

Video tutorial,  Using the simulation  (Investigate A)

Questions

Question

Using the simulation, can you create a situation where the two points are NEVER together?

Video Tutorial   (Investigate B)

Question

Suppose we consider the past,  as if the points had been travelling forever, and our simulation just captures them at a particular starting point.
Does that change your answer to Question 1?

Video Tutorial   (Investigate C)

Video Tutorial   (Investigate D)

Question

Using the simulation, can you create a situation where the two points are ALWAYS together?

Video Tutorial (Investigate E)

Question

Using the simulation, can you create a situation where one point overtakes and passes the other?

Video Tutorial   (Investigate F)

Question

Using the following initial conditions, create a distance-time graph for the motion of the two points,  with times from 0 - 20 s.

Blue Start:	0 cm
Red Start:		5 cm
Blue Speed:	1 cm/s
Red Speed:	1 cm/s

Video Tutorial    (Investigate G)

Question

Using the following initial conditions, create a distance-time graph for the motion of the two points,  with times from 0 - 20 s.

Blue Start:	5 cm
Red Start:		5 cm
Blue Speed:	1 cm/s
Red Speed:	1 cm/s

Video Tutorial    (Investigate H)

Question

Using the following initial conditions, create a distance-time graph for the motion of the two points,  with times from 0 - 20 s.

Blue Start:	5 cm
Red Start:		2 cm
Blue Speed:	1 cm/s
Red Speed:	2 cm/s

Video Tutorial   (Investigate I)

Question

Describe the possible results when the two points have the same speed?

Video Tutorial   (Investigate J)

Question

Describe the possible results when the two points have different speeds?

Video Tutorial   (Investigate K)

Question

What are the connections between the moving points, and the slope-intercept form of an equation?

Video Tutorial   (Investigate L)

Question

Using the online graphing calculator, Desmos,  create pairs of graphs that meet the following criteria:

a)	positive slope and they NEVER MEET

b)	negative slope and they NEVER MEET

c)	two lines that meet at one point and have the same y-intercept

d)	two lines that meet at an infinite number of points and have the same y-intercept

e)	two lines that meet at one point and have different y-intercepts

Video Tutorial   (Investigate M)

Conclusions

There are only three possible ways that we can create a system of two linear equations.

If the slopes of the lines are different,  the two lines WILL meet in a single point.  (one solution)

If the slopes of the lines are the same,  the lines either....
NEVER MEET  (no solution)
or
ALWAYS MEET  (infinite solutions)

Link the Ideas

Link the Ideas summarizes the math in the lesson.
A presentation of ideas connected to the lesson focus.

TEXT Pages 447 - 448

Read/Work through the  Link the Ideas  section on Pages 447 - 448 in your textbook.

Examples

Examples will give you TOOLS for solving problems, and also show you that there may be different ways to approach a question.

Example 1  Connect the Number of Solutions to the Situation

TEXT Pages 448 - 450

Work through Example 1 on Pages 448 - 450 in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Example 2  Predict and Conﬁrm the Number of Solutions

TEXT Pages 451 - 452

Work through Example 2 on Pages 451 - 452 in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Example 3  Identify Zero and Inﬁnite Solutions by Comparing Coefﬁcients

TEXT Page 453

Work through Example 3 on Page 453 in your textbook.

Watch

Video Tutorial

Your Turn

YOUR TURN gives YOU the chance to try to do a problem ON YOUR OWN,  using the example as  a model.

Use the Video Tutorial for help, or to check your work, BUT... you should do your best to do this question ON YOUR OWN FIRST!

Video Tutorial

Key Ideas

Key Ideas summarizes the main new concepts of the lesson.

•	Carefully read through each of the bullet points in this section.

•	If you are unclear about any of these ideas,  ASK YOUR TEACHER FOR HELP!

Check Your Understanding    (Practice Problems)

Doing Practice Problems

Practice problems reinforce the math presented in this lesson.

•	Work through the practice problems for this lesson.
	These problems are for practice, you don’t need to hand them in!
	(your teacher may assign specific questions for you to complete)

•	Use the answers in the back of the book to CHECK YOUR WORK!

•	If you make a mistake, TRY TO FIX IT!

•	If you need help or want more information, view the video solutions provided.

•	If you still need more help,  CONTACT YOUR TEACHER!

Types of Practice Problems

Practice questions require basic understanding of the ideas of the  lesson.

Apply questions will require you to dig deeper for a solution.

Extend questions offer enrichment and extension.

Create Connections questions connect the math to everyday contexts.

Mini Lab (optional)  your teacher may or may not have you do these.

Notes

Question 7,    I made a video,  but... this questions seems pretty far - fetched, even for a math book...

Question 17,    Here is a link to the Desmos page I used to create the video.

Question 19,    I didn’t do a video for this one.
The answer is “no”, lines can only meet in one point OR be the same line and meet everywhere.
There aren’t any other possibilities (unless we are doing geometry on a sphere or other surface)

Question 20,    I didn’t do a video for this one.

Question 21,    I didn’t do a video for this one.

Video Solution Links

Check your work for every problem you do!
If you make a mistake,  try to fix it!
Use the video solutions for more information.
If a problem still doesn’t make sense, ASK YOUR TEACHER FOR HELP!

({{"Question 1", "Question 2", "Question 3", "Question 4", "Question 5"}, {"Question 6", "Question 7", "Question 8", "Question 9", "Question 10"}, {"Question 11", "Question 12", "Question 13", "Question 14", "Question 15"}, {"Question 16", "Question 17", "Question 18",  ,  }, {  , "Question 22", , , }})