# Simultaneous eqns in maple

I am trying to solve the following system of equations in maple but it doesn't work for some reason:

solve({
a*(1-x)-x*f-x*e = 0,
b*(1-x)-x*c-x*d = 0,
c*(1-z)-z*b-z*a = 0,
d*(1-z)-z*e-z*f = 0,
e*(1-y)-y*d-y*c = 0,
f*(1-y)-y*a-y*b = 0,
a+b+c+d+e+f-1 = 0 },
{a, b, c, d, e, f
})

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I tried just now and for me, no answers in outputs, so, no solutions. – Sigur Sep 4 '12 at 2:07

This linear system of equations is inconsistent. One way to see this is to recognize that the first 6 equations imply that the variables a to f all have value zero. But the last equation dictates that their sum is equal to 1. Clearly, if they are all zero then they cannot add up to 1.

eqs:=[a*(1-x)-x*f-x*e = 0,
b*(1-x)-x*c-x*d = 0,
c*(1-z)-z*b-z*a = 0,
d*(1-z)-z*e-z*f = 0,
e*(1-y)-y*d-y*c = 0,
f*(1-y)-y*a-y*b = 0,
a+b+c+d+e+f-1 = 0]:

vars:=[a, b, c, d, e, f]:

with(LinearAlgebra):


Now compare results from

linsys:=GenerateMatrix(eqs[1..6],vars,augmented);
LinearSolve(linsys);

LUDecomposition(GenerateMatrix(eqs[1..6],vars,augmented),output=R);
%[1..-1,1..6].Vector(vars)=%[1..-1,7];


with that from,

linsys:=GenerateMatrix(eqs,vars,augmented);
LinearSolve(linsys);

LUDecomposition(GenerateMatrix(eqs,vars,augmented),output=R);
%[1..-1,1..6].Vector(vars)=%[1..-1,7];

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The equations look linear in $a,b,c,d,e,f$. As such, you can use $$A,b:=LinearAlgebra[GenerateMatrix](system\_of\_equations,variables);$$ and then $LinearAlgebra[LinearSolve](A,b)$ to solve the matrix system.

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