Question 918070
x+y+z=900000,
0.06*x+0.05*y+0.07*z=52000,
x=1.5z

1,1,1,900000
.06,.05,.07,52000
1,0,-1.5,0


900000 52000 0
1 1 1
0.06 0.05 0.07
1 0 -1.5
1st=0.05 0.07 0 -1.5
2nd=0.06 0.07 1 -1.5
3rd=0.06 0.05 1 0
4th=1 1 0 -1.5
5th=1 1 1 -1.5
6th=1 1 1 0
7th=1 1 0.05 0.07
8th=1 1 0.06 0.07
9th=1 1 0.06 0.05
all minors
1=0.05 0.07 0 -1.5
2=0.06 0.07 1 -1.5
3=0.06 0.05 1 0
4=1 1 0 -1.5
5=1 1 1 -1.5
6=1 1 1 0
7=1 1 0.05 0.07
8=1 1 0.06 0.07
9=1 1 0.06 0.05
mult ad
 ad-bc  1*4-2*3
1--
0.05 0.07 0 -1.5
(0.05*-1.5)-(0.07*0)
-0.075-(0.0) =-0.075
2--
0.06 0.07 1 -1.5
(0.06*-1.5)-(0.07*1)
-0.09-(0.07) =-0.16
3--
0.06 0.05 1 0
(0.06*0)-(0.05*1)
0.0-(0.05) =-0.05
4--
1 1 0 -1.5
(1*-1.5)-(1*0)
-1.5-(0) =-1.5
5--
1 1 1 -1.5
(1*-1.5)-(1*1)
-1.5-(1) =-2.5
6--
1 1 1 0
(1*0)-(1*1)
0-(1) =-1
7--
1 1 0.05 0.07
(1*0.07)-(1*0.05)
0.07-(0.05) =0.02
8--
1 1 0.06 0.07
(1*0.07)-(1*0.06)
0.07-(0.06) =0.01
9--
1 1 0.06 0.05
(1*0.05)-(1*0.06)
0.05-(0.06) =-0.01
 matrix of minors
-0.075 -0.16 -0.05
-1.5 -2.5 -1
0.02 0.01 -0.01
original
1 1 1
0.06 0.05 0.07
1 0 -1.5
matrix of cofactors-switching signs
-0.075 same -0.075
-0.16 switch 0.16
-0.05 same -0.05
-1.5 switch 1.5
-2.5 same -2.5
-1 switch 1
0.02 same 0.02
0.01 switch -0.01
-0.01 same -0.01
matrix of cofactors
-0.075 0.16 -0.05
1.5 -2.5 1
0.02 -0.01 -0.01
adjoint transpose
-0.075 1.5 0.02
0.16 -2.5 -0.01
-0.05 1 -0.01
original by row
(row 1)1 1 1
(row 2)0.06 0.05 0.07
(row 3)1 0 -1.5
original by column
(column 1)1 0.06 1
(column 2)1 0.05 0
(column 3)1 0.07 -1.5

1) determinant using  row three
1*0.02=0.02
0*-0.01=0
-1.5*-0.01=0.015
determinant=0.035
and finally the inverse
-0.075/0.035 1.5/0.035 0.02/0.035
0.16/0.035 -2.5/0.035 -0.01/0.035
-0.05/0.035 1/0.035 -0.01/0.035

2) same determinant using column 2
1*0.16=0.16
0.05*-2.5=-0.125
0*-0.01=0
determinant=0.035
 the inverse in fractions
-0.075/0.035 1.5/0.035 0.02/0.035
0.16/0.035 -2.5/0.035 -0.01/0.035
-0.05/0.035 1/0.035 -0.01/0.035
 the inverse in decimals
-2.14285714 42.8571429 0.57142857
4.57142857 -71.4285714 -0.28571429
-1.42857143 28.5714286 -0.28571429
900000 52000 0
column one
-2.14285714 42.8571429 0.57142857
-1928571.43 2228571.43 0.0
solutions
x=300000.0 y= 400000 z= 200000

x = $300000 at 6%
y = $400000 at 5%
z = $200000 at 7%

Here is the gauss jordan method
1,1,1,900000
.06,.05,.07,52000
1,0,-1.5,0



add  down (-0.06) *row 1 to row 2
1,1,1,900000
0,-0.01/1,0.01/1,-2000
1,0,-1.5/1,0

add  down (-1) *row 1 to row 3
1,1,1,900000
0,-0.01/1,0.01/1,-2000
0,-1,-2.5/1,-900000

divide row 2 by -0.01
1,1,1,900000
0,1,0.01/-0.01,-2000/-0.01
0,-1,-2.5/1,-900000

add  down (1) *row 2 to row 3
1,1,1,900000
0,1,0.01/-0.01,2000/0.01
0,0,0.035/-0.01,-700000

divide row 3 by 0.035/-0.01
1,1,1,900000
0,1,0.01/-0.01,2000/0.01
0,0,1,7000.0/0.035

We now have the value for the last variable.
We will work our way up and get the other solutions.

add up  (-0.01/-0.01) *row 3 to row 2
1,1,1,900000
0,1,0,-1.4/-0.0000035
0,0,1,7000.0/0.035

add up  (-1) *row 3 to row 1
1,1,0,700000
0,1,0,1.4/0.0000035
0,0,1,7000.0/0.035

add up  (-1) *row 2 to row 1
1,0,0,1.05/0.0000035
0,1,0,1.4/0.0000035
0,0,1,7000.0/0.035

final
1,0,0,1.05/0.0000035
0,1,0,1.4/0.0000035
0,0,1,7000.0/0.035

1,0,0,1.05/0.0000035 = 300000.0
0,1,0,1.4/0.0000035 = 400000.0
0,0,1,7000.0/0.035 = 200000.0
"1.05/0.0000035","1.4/0.0000035","7000.0/0.035"
(1.05/0.0000035,1.4/0.0000035,7000.0/0.035)
(300000.0,400000.0,200000.0)