Download Gaussian Elimination to solve systems of linear equations

Gaussian Elimination to solve systems of linear equations
Consider 2 simultaneous equations with 2 unknowns:
5x +3y = 11 ………..(1)
2x - 2y = -2……….. (2)
Divide equation (1) by 5, ie
x + 3y = 11 ………..(3)
5
5
multiply equation (3) by 2 and then subtract it from equation (2)
2x - 2y = -2
- 2x + 6y = 22
5
5
0 - 16y = -32 , ie y =2 , and hence x = 1
5
5
The procedure above to solve the equations forms the basis of a general
procedure using a matrix ( a rectangular array of numbers). In this case the
matrix is an augmented matrix, whereby the equation coefficients ( ann) and
constants( kn) are shown ie
[ a11 a12 k1 ] or [ 5 3 11]
[ a21 a22 k2 ]
[ 2 -2 -2 ]
Divide the first row by 5, ie [ 1 3/5 11/5 ]
Now subtract 2x the first row (ie [ 2 6/5 22/5 ] from the second row to give a
zero in the first column of the second row ie
[ 1 3/5 11/5 ]
[ 0 -16/5 -32/5]
Now divide the second row by -16/5 ie [ 0 1 2 ] ie y = 2
To obtain x multiply the second row by -3/16 and subtract from row 1
Ie
[ 1 3/5 11/5 ]
= [ 1 0 1] ie x = 1
- [ 0 3/5
6/5 ]
The use of a matrix to solve 2 simultaneous equations with 2 unknowns is
cumbersome but forms the basis of a general procedure, called Gaussian
Elimination to solve n simultaneous equations with n unknowns
Examples
Use matrices to solve the following equations by elimination
(a) x + y = 3, 2x - y = 3 (b) 4x + y = 7, 5x – y = 2
(c) x + 2y = 1, -x + 2y = 3, (d) 2x - 3y =7, x + y = 1
.
Use Gaussian Elimination to find the values of x, y and z in the simultaneous
equations:
(e) 2x + 3y - 2z = 6, 3x + 2y + 3z = 3, 4x - 5y + 7z = 6
(f) x + 2y + 2z = 7, 2x + y + z = 5, 3x - 2y + 2z = 5