Use Gauss-Jordan elimination method to solve
2x-5y+5z=17
x-2y+3z=9
-x+3y=-4
Line up the terms vertically:
2x - 5y + 5z = 17
x - 2y + 3z = 9
-x + 3y = -4
Fill up the "hole" in the bottom
equation with "+ 0z:
2x - 5y + 5z = 17
x - 2y + 3z = 9
-x + 3y + 0z = -4
Fill in all invisible 1's for
coefficients:
2x - 5y + 5z = 17
1x - 2y + 3z = 9
-1x + 3y + 0z = -4
Erase all the letters and
equal signs:
2 - 5 + 5 17
1 - 2 + 3 9
-1 + 3 + 0 -4
Erase the plus signs
and move the minus
signs close to the
numbers as negative
signs:
2 -5 5 17
1 -2 3 9
-1 3 0 -4
Draw a vertical line
where the equal signs
were and put brackets
around the whole thing.
[ 2 -5 5 | 17]
[ 1 -2 3 | 9]
[-1 3 0 | -4]
This is called the
augmented matrix
The idea is to get 0's
in the lower left three
positions, where the
three red numbers are
below:
[ 2 -5 5 | 17]
[ 1 -2 3 | 9]
[-1 3 0 | -4]
To get a 0 where the 1 is,
temporarily multiply the top
row thru by 1 and the middle
row thru by -2.
1[ 2 -5 5 | 17]
-2[ 1 -2 3 | 9]
[-1 3 0 | -4]
[ 2 -5 5 | 17]
[-2 4 -6 |-18]
[-1 3 0 | -4]
Add the top row to the middle
row and leave the top row as
it is:
[ 2 -5 5 | 17]
[ 0 -1 -1 | -1]
[-1 3 0 | -4]
To get a 0 where the -1 at the
bottom left is, multiply the
top row by 1 and the bottom row
by 2:
1[ 2 -5 5 | 17]
[ 0 -1 -1 | -1]
2[-1 3 0 | -4]
[ 2 -5 5 | 17]
[ 0 -1 -1 | -1]
[-2 6 0 | -8]
Add the top row to the bottom
row, leaving the top row as it
is:
[ 2 -5 5 | 17]
[ 0 -1 -1 | -1]
[ 0 1 5 | 9]
To get a 0 where the 1 is,
multiply the middle row by 1
and add to the bottom row:
[ 2 -5 5 | 17]
1[ 0 -1 -1 | -1]
1[ 0 1 5 | 9]
[ 2 -5 5 | 17]
[ 0 -1 -1 | -1]
[ 0 0 4 | 8]
Now that there are 0's
in those three positions,
we rewrite the augmented
matrix as a system of
equations, by putting the
variables and equal signs
back in:
[ 2x -5y 5z = 17]
[ 0x -1y -1z = -1]
[ 0x 0y 4z = 8]
Erase the brackets, the
terms with 0 coefficients,
the 1's, and move the negative signs
left as minus signs:
2x - 5y + 5z = 17
-y - z = -1
4z = 8
Solve the bottom equation for z:
4z = 8
z = 2
Substitute z = 2 into the middle
equation:
-y - z = -1
-y - 2 = -1
-y = 1
y = -1
Substitute y = -1 and z = 2 into
the top equation:
2x - 5y + 5z = 17
2x - 5(-1) + 5(2) = 17
2x + 5 + 10 = 17
2x + 15 = 17
2x = 2
x = 1
So the solution is
(x, y, z) = (1, -1, 2)
Edwin
