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### Example:

 2 x +3 y +7 z = 2 x -    y -   z = 1 x +  z = 1

A =

 2 3 7 1 -1 1 1 0 1

V =

 2 1 1

STANDARD METHOD:

GAUSSIAN ELIMINATION:

1. Solve first equation for x by dividing it by two and moving y and z terms to other side:

x + (3/2) y + (7/2) z = 1

x = 1 - (3/2) y - (7/2) z

1.Divide first row of A and V by 2:

A =

 1 3/2 7/2 1 -1 1 1 0 1

V =

 1 1 1

2. Substitute in other equations for x:

-5/2  y  - 9/2 z = 0, for second equation.
- 3/2 y  - 5/2 z = 0, for third equation.

2. Subtract first row of A and V from each other row of A and V:

A =

 1 3/2 7/2 0 -5/2 -9/2 0 -3/2 -5/2

V =

 1 0 0

3. Solve second equation for y:

y = -9/5 z

3. Divide second row by -5/2:

A =

 1 3/2 7/2 0 1 9/5 0 -3/2 -5/2

V =

 1 0 0

4. Substitute for y in third:

1/5 z = 0

4. Add 3/2 of second row to third:

A =

 1 3/2 7/2 0 1 9/5 0 0 1/5

V =

 1 0 0

5. Multiply third equation by 5:

z = 0

5. Multiply third row by 5:

A =

 1 3/2 7/2 0 1 9/5 0 0 1

V =

 1 0 0

Current form of equations:

x = 1 - (3/2) y - (7/2) z
y = - 9/5 z
z = 0

Current A and V:

A =

 1 3/2 7/2 0 1 9/5 0 0 1

V =

 1 0 0

6. Substitute for z in first two equations:

x = 1 - (3/2) y
y = 0
z = 0

6. Subtract 7/2 of third row from first and 9/5 of it from second row:

A =

 1 3/2 0 0 1 0 0 0 1

V =

 1 0 0

7. Substitute for y in the first equation:

x = 1
y = 0
z = 0

7. Subtract 3/2 of 2nd row from 1st:

A =

 1 0 0 0 1 0 0 0 1

V =

 1 0 0

You see, if you follow this example, that solving equations by systematically eliminating variables, and using Gaussian elimination to change a matrix into the identity matrix are essentially the same thing.