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Simultaneous Linear Equations: Substitution and Elimination

Introducing Algebraic Solutions of Simultaneous Equations to Year 10 (Turning 16 this year)
by

Darryl Monahan

on 2 December 2013

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Transcript of Simultaneous Linear Equations: Substitution and Elimination

Year 10
SIMULTANEOUS EQUATIONS
photo credit Nasa / Goddard Space Flight Center / Reto Stöckli
How old are Xavier and Yvonne?
Y
vonne is twice as old as
X
avier
, and
WE COULD USE "GUESS AND CHECK"
a.k.a. "trial and error"
We need
more elegant
and

reliable

methods
of solving these
kinds of problems ...
BUT this is
not
going to work when the situation becomes more complex!
...if we don't want to be forced to go
back to school
as adults.
Y
vonne is twice as old as
X
avier
, and
Y
vonne is also 3 years older than
X
avier.
BACK TO OUR FUTURE...
How old are Xavier and Yvonne?
If we let
x
= Xavier's age
and let
y
= Yvonne's age

then, we can complete
Step 1
...
STEP 1: TRANSLATE INFORMATION

into math language
Since Yvonne (y) is
twice as old
as Xavier (x),
we can say:

y
= 2
x
Now we have at least four ways available to
solve
these
simultaneous equations
:

1 Trial and Error
2
Graphically
3 By
Substitution
4 By
Elimination
Graphical Solution to Simultaneous Equations
Stop and explore the contents of the following
SATELLITE STATIONS...
Click a content item in a station to zoom in and the back arrow to zoom out, OR pan and use your own zoom.
To Explore:
Solving by SUBSTITUTION
by D.Monahan
Chapter 4A
Chapter 4B
Solving by ELIMINATION
Chapter 3F
basic
intermediate
advanced
Launchpad
One equation with one variable can be solved by
transposition
...
if two unknowns are involved
, we
need a second equation
involving them, and the solution to these
is not often straight-forward...
Simultaneous Equations
Transposing Equations Revision
AVAILABLE Chapter 2D & 2E or click link for a
Powerpoint
how-to...
BUT
However, let's keep it as simple as possible for now...
Self-paced, differentiated sub-unit of Year 10 LINEAR FUNCTIONS UNIT with Distance-Time context emphasis
about
Substitution Practice Range
basic
intermediate
advanced
Elimination Practice Range
basic
intermediate
advanced
Prerequiste Knowledge & Skills
Chapter 2A to 2D
or 2E
Graphical Solutions
Solving by Substitution
Solving by Elimination
Revision: Chapter 2A
Substituting into Equations
More Practice? Exercise 3E
More Practice? Exercise 3F
learning objectives
Room for Improvement
1. Nest Prezi in Power-point, so animated power-point slides could be used, as well as Prezi.
2. Include some flash animation
3. Improve PPT Slide Content
5. Explore Prezi Add-ons that allow sub-paths in Prezi
After initial introduction to Navigation (Path and Sandpit) in Prezi, Students download a copy or are given viewing rights and explore
Y
vonne is also 3 years older than
X
avier.
These
facts
about Yvonne and Xavier's ages are
true
simultaneously
(i.e. at the same time),
so these
equations are true simultaneously
And, since Yvonne is
3 years older
than Xavier,
we can say:

y
=
x
+3
Applying what you know so far
Intermediate
For ADVANCED Graphical Problems, work through Chapter 4E, 4F & 4G
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/3/Distance%20vs%20Time.%20Linear%20Algebra%20+%20Functions/Substitution%20Worksheet.Basic.docx
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/3/Distance%20vs%20Time.%20Linear%20Algebra%20+%20Functions/Substitution%20Worksheet.Intermediate.docx
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/3/Distance%20vs%20Time.%20Linear%20Algebra%20+%20Functions/Substitution%20Worksheet.Advanced.docx
NB: All hyper-linked documents are stored at end of the following path on qportal:
Home > Student Workspaces > Year 10 > Maths > 05-Calculator > Documents Hyper-linked to Year 10 Simultaneous Eqns PREZI
4. Improve Worksheets in Practice Range - perhaps more Distance-Time orientated
Transposition SkillSheet
then try
Finding x and y intercepts
DOWNLOAD A COPY
https://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/SkillSHEET%203.2%20Intercepts.pdf
DOWNLOAD A COPY:
Plain Graph Paper
Download a Copy:
Factorising Quadratics Revision
You might want to brush up on this first!
Powerpoint Talkie
or just check this out:
basic
intermediate
advanced
6. Add Ti-Nspire examples of Solving Sim Eqns
https://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/Elimination%20Worksheet.Basic.Word.docx
Steve (the monkey) says...
End of the line...
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/Transposing%20Equations.pptx
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/01Z%20Factorising%20Quadratics.ppsx
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/SkillSHEET%204.3%20TRANSPOSING%20EQUATIONS.pdf
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/Rules%20Graphs%20and%20Substitution%20TASK%20Word.docx
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/plain%20graph%20paper.pdf
Class Activity
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/Revisit%20Taking%20Photographs%20From%20a%20Helicopter-%20Intro%20to%20sub%20method%20CLASS%20ACTIVITY.doc
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/Elimination%20Worksheet.Intermediate.docx
Distance/Velocity/Acceleration
NOT FOR THE FAINT HEARTED
COMING SOON...
1. Estimating Velocity by drawing tangents to Distance-Time curves and finding their gradients.
2. Similarly for Acceleration from Velocity-Time Graphs
3. Link to Linemotion Applet
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/linemotion.zip
Solve as best you can...
Two wombats start at the same time on opposite sides of a rectangular paddock. One is faster than the other. They cross at a point 720 metres from the left side on their way to the other end. When each reaches the other side it stops and eats for 11.5 minutes. The wombats meet again at a point 400 metres from the right side of the paddock.

If p = speed of the wombat starting from left,
q = speed of the wombat starting from right, and
x = width of the paddock

How wide is the paddock?
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/Elimination%20Worksheet.Advanced.docx
PIGS IN SPACE
Or, test your skills at
solving basic equations
basic
intermediate
advanced
The different uses of
the Cartesian Plane
Test your understanding
http://www.sycd.co.uk/dtg/
http://qportal.kew.vic.edu.au/virtualclass/y10/maths/5/Documents%20Hyper-linked%20to%20Year%2010%20Simultaneous%20Eqns%20PREZI/xy%20vs%20dt%20graphs.docx
Link to WORD Doc form
ANSWERS
You will know what SIMULTANEOUS EQUATIONS are
You will be reminded how to solve them graphically
You will know TWO ways to solve them algebraically
Full transcript