Solving%20System%20Of%20Linear%20Equations

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Solving System Of Linear Equations ?????
???????
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Solve the following system of linear equations
1.
Elimination????? Substitution?????
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2.
Gaussian Elimination ???
z 1, y 1, x 4
Alternative
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Alternative
Elementary row transformations

• Interchanging 2 rows Ri ?Rj
• Multiply a row by a constant Ri ?kRi
• Add to a row a multiple of another row Ri ?kRjRi

R2 ? 2R1 R2 R3 ? 3R1 R2
R3 ? R2 R3
R3 ?0.5R3
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ALGORITHM
R1 ? 3R3 R1 R2 ? 2R3 R2
R1 ? 2R2 R1
The matrix can be simplified further
x 4, y 1, z 1
Computer Simulation
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PRACTICE
1.
2.
In general, how many solution(s) can a system of
linear equations have?
3.
Solution
BACK
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SOLUTION
BACK
0z2 no solution
x (1?)/2, y 4x 3 z
2 ? where ??IR
z?,
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Computer program
1. Solving linear equations in 3 variables

• http//www.mkaz.com/math/js_lalg3.html

2. Step by Step Illustration
All steps shown http//www1.minn.net/dchristo/
GElim/GElim.html With final score
http//wims.unice.fr/wims/wims.cgi?langcncmdnew
moduleU12Falgebra2Fvisgauss.cntypesystemsiz
e3fieldQ
Further points of discussion

• How can the ALGORITHM described above be applied
  to
• 4 linear equations in 4 variables?
• n linear equations in n variables?

2. What are the limitations of the above
computer programs?
Websites http//cos.cumt.edu.cn/math/shuxuekejian/
xianxingdaishu/charp1/124.htm
http//ws1.hkcampus.net/ws1-kcy/al_pm
ath.html
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A Mathematical Problem
Amy, Ben and Calvin play a game as follows. The
player who loses each round must give each of the
other players as much money as the player has at
that time. In round 1, Amy loses and gives Ben
and Calvin as much money as they each have. In
round 2, Ben loses, and gives Amy and Calvin as
much money as they each then have. Calvin loses
in round 3 and gives Amy and Ben as much money
as they each have. They decide to quit at this
point and discover that they each have 24. How
much money did they each start with?
Approach (1) top down strategy (2) bottom up
strategy
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Method 1 Top down strategy
Let Amy, Ben and Calvin each had x, y and z
initially.
Amy Ben Calvin
At start x y z
After round 1
After round 2
After round 3
3y x z
7z x y
x y z 6, 3y x z 12 and 7z x y
24 from which x, y and z can be solved.
x39, y21, z12
NEXT
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Method 2 Bottom up strategy
Instead of considering how much money Amy, Ben
and Calvin had originally, work backwards from
the moment when each of them has 24 each.
Complete the following table and see how easily
you can reach exactly the same conclusion as in
method 1.
Amy Ben Calvin
At start x y z
After round 1
After round 2
After round 3
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