underdetermined system I have a big trouble while solving a problem with is called underdetermined system, which have a form:
$$a_1x_1 + b_1x_2 + c_1x_3 + d_1x_4 + e_1x_5 + f_1x_6 + g_1x_7 = n_1\\
a_2x_1 + b_2x_2 + c_2x_3 + d_2x_4 + e_2x_5 + f_2x_6 + g_2x_7 = n_2\\
a_3x_1 + b_3x_2 + c_3x_3 + d_3x_4 + e_3x_5 + f_3x_6 + g_3x_7 = n_3\\$$
Where $a,b,c,d,e,f,g,n$ are constants. There are 3 equations with 7 unknow.
I would like to solve this system to find $x_1,x_2,...x_7$ with constraint that $sum(x_1,x_2,...x_7) = 1$.
I would greatly appreciate if you help me.
Thanks in advance
 A: I propose a simpler example in order to understand what happens with this kind of undetermined systems. imagine you have one equation and three variables, for example
$$x-y+z=2$$
This system is clearly undetermined, because you cannot solve $x,y$ and $z$ variables. However, you can make easily some solutions by choosing for different values of the variables. For example, let's take $x=1,y=0$ then, it follows that $z=1$. Then $(1,0,1)$ is a solution of the system. Or we can take $x=0,y=1$, following that $z=3$. then, another solution is $(0,1,3)$.
As you get only one independent equation and three variables, there will be two linearly independent solutions. In this case these independent solutions can be the ones we got $(1,0,1)$ and $(0,1,3)$. then, all the solutions can be written as linear combinations of these two. Meaning that, for any value of $m,n$
$$(x,y,z)=m(1,0,1)+n(0,1,3)$$
will be always a solution for the system. Mathematically, it can be expressed as $\mbox{lin}[(1,0,1),(0,1,3)]$ (if you know a little bit of linear algebra, you will be familiar with this '$\mbox{lin}$' for generating linear spaces).
In your system, there are 4 equations and 7 variables, so you have to find 3 linearly independent solutions. It is a little bit more difficult, but the concepts are the same. I hope it will be useful for you!
A: Does it look better for you if I write $$a_1x_1 + b_1x_2 + c_1x_3 + d_1x_4 = n_1- e_1x_5 - f_1x_6 - g_1x_7=A_1$$
$$a_2x_1 + b_2x_2 + c_2x_3 + d_2x_4  = n_2- e_2x_5 - f_2x_6 - g_2x_7=A_2$$
$$a_3x_1 + b_3x_2 + c_3x_3 + d_3x_4  = n_3- e_3x_5 - f_3x_6 - g_3x_7=A_3$$
 $$x_1+x_2+x_3+x_4=1-x_5-x_6-x_7=A_4$$
