How to solve a system of equations (100 in number) using a program?

Question

Form: $x_i+x_j-x_k-C_i=0$

To clarify, it might have been better to write this here as: $x_i+x_j-y_k-C_i=0$, since $x_i$ and $x_j$ come from a single set of unknowns. $y_k$ are considered different.

All are constants (but dependent mostly) but only $C_i$ are known. The problem is, it's not in matrix form. I have a set of equations, n in number, with all the different $x_k$ being exactly n in number, while $x_i$ and $x_j$ added together are fewer than n in number. I want to elucidate the dependent variables, but doing it by hand takes too long. How to express this as input to a (preferably free) pc program, to get solution e.g., as an upper triangular matrix?

Update: This seems easy at the moment (although I haven't tested yet whether this is correct), but the format of the problem made getting here difficult: I need to make a mostly sparse matrix with +/-1's for the three types of unknowns. One side would be as long as the sum of their indices, an additional column for $C_i$, and the number of rows equal to the number of equations. If this is correct, then what remains is either to find a text parser to convert plain text equations to matrix form, or to just fill in the matrix by hand.

$A$, in sparse format (row, col, value):

1 1 1
2 1 1
3 2 1
4 3 1
5 3 1
6 4 1
7 4 1
8 4 1
9 5 1
10 5 1
11 6 1
12 6 1
13 7 1
14 9 1
15 9 1
16 10 1
17 11 1
18 12 1
19 12 1
20 13 1
21 13 1
22 14 1
23 14 1
24 14 1
25 15 1
26 15 1
27 16 1
28 17 1
29 17 1
30 17 1
31 18 1
32 19 1
33 20 1
34 20 1

Coefficients:

30
27
26
26
24
25
25
20
17
21
13
14
17
18
17
13
14
13
12
12
11
6
2
3
3
2
4
2
3
0
2
1
4
4

Answer 1

Ok, I assume you know how to convert a set of linear equations into matrix form (if not, please check out this wikipedia page). So, what you have is a system of the form: $$ A \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_m \\ y_1 \\ y_2 \\ \vdots \\ y_n \end{pmatrix} = \begin{pmatrix} C_1 \\ C_2 \\ \vdots \\ C_n \end{pmatrix} \tag{1} $$ where $A$ is an $n \times (m+n)$ matrix with entries: $\{ 0, +1, -1 \}.$ The system $(1)$ is underdetermined linear system, i.e., you have more variables than the number of equations. In general, there is not unique solution to this system.

Among the infinitely many solution vectors $v$ to $Av = C,$ you should decided a criteria to narrow down which $v$ you are looking for. If you are looking for sparsest $v$ then you can solve this problem with $\ell_1$ minimization: $$ \text{min } \| v \|_1 \text{ subject to } Av = C.$$ or if you're looking for shortest $v$ in the Euclidean sense, then you can solve the the $\ell_2$ minimization problem: $$ \text{min } \| v \|_2 \text{ subject to } Av = C.$$

Either ways, it's a convex optimization problem solvable in about $\mathcal O(n^2m).$