Find x,y,z, and w by the GAUSE- JORDAN method?

Question

5x+3y+z+5w=-2
3x+2y+2z+4w=2
2x+4y+3z-3w=-11
4x-3y-2z+2w=-3

Answer 1

5x+3y+z+5w=-2
3x+2y+2z+4w=2
2x+4y+3z-3w=-11
4x-3y-2z+2w=-3

=>
5 3 1 5 I -2
3 2 2 4 I 2
2 4 3 -3 I -11
4 -3 -2 2 I -3

R4 - (4/5)*R1 , R3 - (2/5)*R1 and R2 - (3/5)*R1 will give

5.0000 3.0000 1.0000 5.0000 -2.0000
0 0.2000 1.4000 1.0000 3.2000
0 2.8000 2.6000 -5.0000 -10.2000
0 -5.4000 -2.8000 -2.0000 -1.4000

R4 - (-5.4/0.2)*R2 , R3 - (-5.2/0.2)*R2 will give

5.0000 3.0000 1.0000 5.0000 -2.0000
0 0.2000 1.4000 1.0000 3.2000
0 0.0000 -17.0000 -19.0000 -55.0000
0 0.0000 35.0000 25.0000 85.0000

R4 - (-37/17)R3 will give

5.0000 3.0000 1.0000 5.0000 -2.0000
0 0.2000 1.4000 1.0000 3.2000
0 0.0000 -17.0000 -19.0000 -55.0000
0 0.0000 0.0000 -14.1176 -28.2353

R4 / -14.1176 will give

5.0000 3.0000 1.0000 5.0000 -2.0000
0 0.2000 1.4000 1.0000 3.2000
0 0.0000 -17.0000 -19.0000 -55.0000
0 0.0000 0.0000 1.0000 2.0000

R3 - (-19)R4 will give

5.0000 3.0000 1.0000 5.0000 -2.0000
0 0.2000 1.4000 1.0000 3.2000
0 -0.0000 -17.0000 0 -17.0000
0 0.0000 -0.0000 1.0000 2.0000

R3 /(-17) will give

5.0000 3.0000 1.0000 5.0000 -2.0000
0 0.2000 1.4000 1.0000 3.2000
0 0.0000 1.0000 0.0000 1.0000
0 0.0000 0.0000 1.0000 2.0000

R2- R4 will give

5.0000 3.0000 1.0000 5.0000 -2.0000
0 0.2000 1.4000 0 1.2000
0 0.0000 1.0000 0 1.0000
0 0.0000 0.0000 1.0000 2.0000

R2- (1.4)R3 will give

5.0000 3.0000 1.0000 5.0000 -2.0000
0 0.2000 0.0000 0 -0.2000
0 0.0000 1.0000 0 1.0000
0 0.0000 0.0000 1.0000 2.0000

R3 /(0.2) will give

5.0000 3.0000 1.0000 5.0000 -2.0000
0 1.0000 0.0000 0 -1.0000
0 0.0000 1.0000 0 1.0000
0 0.0000 0.0000 1.0000 2.0000

R1- 5R4 will give

5.0000 3.0000 1.0000 0 -12.0000
0 1.0000 0.0000 0 -1.0000
0 0.0000 1.0000 0 1.0000
0 0.0000 -0.0000 1.0000 2.0000

R1-R3 will give

5.0000 3.0000 0 0 -13.0000
0 1.0000 0.0000 0 -1.0000
0 0.0000 1.0000 0 1.0000
0 0.0000 -0.0000 1.0000 2.0000

R1-3R2 will give

5.0000 0 -0.0000 0 -10.0000
0 1.0000 0.0000 0 -1.0000
0 0.0000 1.0000 0 1.0000
0 0.0000 -0.0000 1.0000 2.0000

R1/5 will give
1.0000 0 -0.0000 0 -2.0000
0 1.0000 0.0000 0 -1.0000
0 0.0000 1.0000 0 1.0000
0 0.0000 -0.0000 1.0000 2.0000

And thats the final answer.

So, your answer is x=-2, y=-1,z=1, w=2

Verification
a=
5 3 1 5
3 2 2 4
2 4 3 -3
4 -3 -2 2
b=
-2
2
-11
-3
inverse of a is

0.0250 -0.0250 0.1250 0.1750
0.4000 -0.4000 0.0000 -0.2000
-0.5125 0.6792 0.1042 0.0792
0.0375 0.1292 -0.1458 -0.0708

Thus,
inv(a)*b = [-2 -1 1 2]'

I hope i answered your question.

All the best

Answer 2

You really should spell Gauss' name correctly.

The GJ method just means to use use the first equation to eliminate (zero out) the x term in eqns 2 on. Then use equation 2 to eliminate coefficients of the y in all other eqns, use eqn 3 to zero out coeffiecients of z in other eqns, etc...

When you run out of eqns or variables, then you're done. Note that you need to use the 'augmented matrix' which includes the vector on the right side.



It amounts to elementary row operations.