Question 251271
<pre><font size = 4 color = "indigo"><b>
I think you must have mistyped the equation 

1+3c+2d+2f=110

as none of the others had a number in front
that you had to subtract from both sides, so
if the answer I get is not the right one it's
because you mistyped that 1 in front of the 
fourth equation

a+b+c+d+e+f=0
a-2b=0
a+2b+3d+2e=220
1+3c+2d+2f=110
a+2b+4d+e+5f=240
a+2d+3e+4f=165 

1a + 1b + 1c + 1d + 1e + 1f =   0
1a - 2b + 0c + 0d + 0e + 0f =   0
1a + 2b + 0c + 3d + 2e + 0f = 220
0a + 0b + 3c + 2d + 0e + 2f = 110-1
1a + 2b + 0c + 4d + 1e + 5f = 240
1a + 0b + 0c + 2d + 3e + 4f = 165

Simplifying the equation that I think
you mistyped:

1a + 1b + 1c + 1d + 1e + 1f =   0
1a - 2b + 0c + 0d + 0e + 0f =   0
1a + 2b + 0c + 3d + 2e + 0f = 220
0a + 0b + 3c + 2d + 0e + 2f = 109
1a + 2b + 0c + 4d + 1e + 5f = 240
1a + 0b + 0c + 2d + 3e + 4f = 165 

This is the matrix AX=B form:

{{{(matrix(6,6,
1,  1,  1,  1,  1,  1, 
1, -2,  0,  0,  0,  0, 
1,  2,  0,  3,  2,  0, 
0,  0,  3,  2,  0,  2, 
1,  2,  0,  4,  1,  5, 
1,  0,  0,  2,  3,  4))*(matrix(6,1,a,b,c,d,e,f))=(matrix(6,1,0,0,220,109,240,165))}}} 

The inverse of that matrix is:

{{{(matrix(6,6,
270/341,    13/32,  -38/341, -90/341, 46/341, -80/341,
135/341,    -9/31,  -19/341, -45/341, 23/341, -40/341,
92/341,     -2/31,  -18/341,  83/341,-50/341, -2/341,
-168/341,    5/31,  107/341,  56/341, 32/341, -26/341,
-18/341,    -5/31,   48/341,   6/341, -94/341, 119/341,
30/341,     -2/31,  -80/341, -10/341, 43/341,  29/341))}}}

We multiply both sides of the {{{AX=B}}} equation by that
inverse matrix, getting it in the form {{{A^(-1)(AX)=A^(-1)B}}}

{{{(matrix(6,6,
270/341,    13/32,  -38/341, -90/341, 46/341, -80/341,
135/341,    -9/31,  -19/341, -45/341, 23/341, -40/341,
92/341,     -2/31,  -18/341,  83/341,-50/341, -2/341,
-168/341,    5/31,  107/341,  56/341, 32/341, -26/341,
-18/341,    -5/31,   48/341,   6/341, -94/341, 119/341,
30/341,     -2/31,  -80/341, -10/341, 43/341,  29/341))*

((matrix(6,6,
1,  1,  1,  1,  1,  1, 
1, -2,  0,  0,  0,  0, 
1,  2,  0,  3,  2,  0, 
0,  0,  3,  2,  0,  2, 
1,  2,  0,  4,  1,  5, 
1,  0,  0,  2,  3,  4))*(matrix(6,1,a,b,c,d,e,f)))=


(matrix(6,6,
270/341,    13/32,  -38/341, -90/341, 46/341, -80/341,
135/341,    -9/31,  -19/341, -45/341, 23/341, -40/341,
92/341,     -2/31,  -18/341,  83/341,-50/341, -2/341,
-168/341,    5/31,  107/341,  56/341, 32/341, -26/341,
-18/341,    -5/31,   48/341,   6/341, -94/341, 119/341,
30/341,     -2/31,  -80/341, -10/341, 43/341,  29/341))*(matrix(6,1,0,0,220,109,240,165))}}} 

Matrix multiplication. though not commutative, is
associative, so we move the parentheses and change the
{{{A^(-1)(AX)=A^(-1)B}}} form to the
{{{(A^(-1)A)X=A^(-1)B}}} form

{{{((matrix(6,6,
270/341,    13/32,  -38/341, -90/341, 46/341, -80/341,
135/341,    -9/31,  -19/341, -45/341, 23/341, -40/341,
92/341,     -2/31,  -18/341,  83/341,-50/341, -2/341,
-168/341,    5/31,  107/341,  56/341, 32/341, -26/341,
-18/341,    -5/31,   48/341,   6/341, -94/341, 119/341,
30/341,     -2/31,  -80/341, -10/341, 43/341,  29/341))*

(matrix(6,6,
1,  1,  1,  1,  1,  1, 
1, -2,  0,  0,  0,  0, 
1,  2,  0,  3,  2,  0, 
0,  0,  3,  2,  0,  2, 
1,  2,  0,  4,  1,  5, 
1,  0,  0,  2,  3,  4)))*(matrix(6,1,a,b,c,d,e,f))=


(matrix(6,6,
270/341,    13/32,  -38/341, -90/341, 46/341, -80/341,
135/341,    -9/31,  -19/341, -45/341, 23/341, -40/341,
92/341,     -2/31,  -18/341,  83/341,-50/341, -2/341,
-168/341,    5/31,  107/341,  56/341, 32/341, -26/341,
-18/341,    -5/31,   48/341,   6/341, -94/341, 119/341,
30/341,     -2/31,  -80/341, -10/341, 43/341,  29/341))*(matrix(6,1,0,0,220,109,240,165))}}}

Next we do the matrix multiplication on both sides, and
since {{{A^(-1)*A=I}}} we have the {{{IX=A^(-1)B}}} form:

{{{

(matrix(6,6,
1,  0,  0,  0,  0,  0, 
0,  1,  0,  0,  0,  0, 
0,  0,  1,  0,  0,  0, 
0,  0,  0,  1,  0,  0, 
0,  0,  0,  0,  1,  0, 
0,  0,  0,  0,  0,  1))*(matrix(6,1,a,b,c,d,e,f))=

(matrix(6,1,
-20330/341, -10165/341, -7243/341, 33034/341, 8289/341,-3585/341))

}}}

When we make the multiplication on the left, we
have the final solution, the {{{X=A^(-1)B}}} form:

{{{(matrix(6,1,a,b,c,d,e,f))=

(matrix(6,1,
-20330/341, -10165/341, -7243/341, 33034/341, 8289/341,-3585/341))

}}}

The answer you wanted is probably all different numbers since you 
probably mistyped that 1 in front of the 4th equation instead of
what it should have been.

I did the above on a TI-84 calculator.

Edwin</pre>