Solve Systems of Equations using Gauss Elimination

Gauss Elimination:

The objective of Gauss elimination is to use a systematic of the usual elimination operations (addition / subtraction of equations, multiplying / dividing equations by a constant) to obtain a "triangularized form" for the system with "zeros" below the diagonal and "ones" on the diagonal:

eg.)
(1) x + 2   y -  3   z                                                                                        (1)
0   x + (1) y -  6   z = 3                                                                                 (2)
0   x +  0   y + (1) z = 2                                                                                (3)

Note:
Zeros below...
"1" is along so-called diagonal.

One this is done, "z" can be found directly from equation (3). Back substitution in eqn. (2) yields "y" then a final back sub. in (1) yields "x"

Example:
Original system:
4 x + 8 y + 4 z = 80
2 x + 1 y  - 4 z = 7
3 x  - 1 y + 2 z = 22

Equivalent System after Gauss Elimination:
1 x + 2 y +    z = 20
0 x + 1 y + 2 z = 11
0 x + 0 y + 1 z = 3

==> x = 7 , y = 5 , z = 3
We will now study Gauss Elimination by way of example.

Gauss Elimination - basic idea:

• Start with 2x2 system:

a₁ x  + b₁ y = c₁
a₂ x + b₂ y = c₂

Steps to achieving "Gauss Form" ... 2x2

Step 1: make a1 = 1		=
Step 2: make a2 = 0	Step 3: make b2 = 1	=

after Gauss Elimination ==>

Gauss Form
1 x + # y = #
0 x + 1 y = #
"zeros" below and "ones" on the diagonal...

Further Elimination ==>
Gauss-Jordan Form
1 x + 0 y = #
0 x + 1 y = #
"zeros" below and above the diagonal and "ones" on the diagonal.

• Start with 3x3 system:

a₁ x + b₁ y + c₁ z = d₁
a₂ x + b₂ y + c₂ z = d₂
a₃ x + b₃ y + c₃ z = d₃

Steps to achieving "Gauss Elimination" ... 3x3

Step 1: make a1 = 1			=
Step 2: make a2 = 0	Step 4: make b2 = 1		=
Step 3: make a2 = 0	Step 5: make b2 = 1	Step 6: make b3 = 1	=

Gauss Elimination ==>
Gauss Form
1 x + # y + # z = #
0 x + 1 y + # z = #
0 x + 0 y + 1 z = #
"zeros" below and "ones" on the diagonal...

Further Elimination ==>
Gauss-Jordan Form
1 x + 0 y + 0 z = #
0 x + 1 y + 0 z = #
0 x + 0 y + 1 z = #
"zeros" below and above the diagonal and "ones" on the diagonal.

Gauss Elimination using Augmented Matrix
Writing the system of equations over and over again can get a bit tedious.
A short hand is available in which we represent a system of equations by a rectangular array of numbers called an augmented matrix.
eg.
2 x + 4 y + 6 z = 4
1 x + 5 y + 9 z = 2
2 x + 1 y + 3 z = 7

can be written in augmented matrix form as:

2	4	6	4
1	5	9	2
2	1	3	7

Example 1 - Solve 2x2 Systems of Equations using Gauss Elimination
Example 2 - Solve 2x2 Systems of Equations using Gauss Elimination
Example 3 - Solve 3x3 Systems of Equations using Gauss Elimination
Example 4 - Solve 3x3 Systems of Equations using Gauss Elimination
Example 5 - Solve 3x3 Systems of Equations using Gauss Elimination (Augmented Matrix)