How to add extra constraints to an equation？

Question

Background：

I have an equation which looks like as follows:

$W \times P = R$

$\left[\begin{array} &{1}&{0}&{0}&-\frac{w_{1}}{w_{o1}} &\dots &{0} &-\frac{w_{1}}{w_{0} } \\{0}&{-\frac{w_{2}}{w_{o2}}}&{1}&{0}&\dots &{0} &-\frac{w_{2}}{w_{o2}} \\ &&&& {\vdots} \\ {1} & {c-1} & {0} & {0} &{\cdots} & {0} & {0} \\ {0} & {0} & {1} & {c-1} & {\cdots} & {0} &{0} \\ &&&&{\vdots}\end{array}\right] \left[ \begin{array}{c}{p_{11}} \\ {p_{1o}} \\ {p_{22}} \\ {p_{2o}} \\ {\vdots} \\ {p_{c c}} \\ {p_{co}}\end{array}\right] = \left[ \begin{array}{l}{0} \\ {0} \\ {\vdots} \\ {0} \\ {1} \\ {\vdots} \\ {1} \end{array}\right]$

c denotes the number of classes and here, c = 5.
The elements of P denote the probability belonging to each class.

For example, $p_{11}$ denotes the probability the 1st sample belongs to the first class. $p_{1o}$ denotes the probability the 1st sample belongs to the classes except the first class.

W = array([[  1.        ,   0.        ,   0.        ,  -1.76690464,
          0.        ,  -1.76690464,   0.        ,  -1.76690464,
          0.        ,  -1.76690464],
       [  0.        , -38.43501272,   1.        ,   0.        ,
          0.        , -38.43501272,   0.        , -38.43501272,
          0.        , -38.43501272],
       [  0.        , -41.64051053,   0.        , -41.64051053,
          1.        ,   0.        ,   0.        , -41.64051053,
          0.        , -41.64051053],
       [  0.        ,  -1.06855322,   0.        ,  -1.06855322,
          0.        ,  -1.06855322,   1.        ,   0.        ,
          0.        ,  -1.06855322],
       [  0.        ,  -2.86308364,   0.        ,  -2.86308364,
          0.        ,  -2.86308364,   0.        ,  -2.86308364,
          1.        ,   0.        ],
       [  1.        ,   4.        ,   0.        ,   0.        ,
          0.        ,   0.        ,   0.        ,   0.        ,
          0.        ,   0.        ],
       [  0.        ,   0.        ,   1.        ,   4.        ,
          0.        ,   0.        ,   0.        ,   0.        ,
          0.        ,   0.        ],
       [  0.        ,   0.        ,   0.        ,   0.        ,
          1.        ,   4.        ,   0.        ,   0.        ,
          0.        ,   0.        ],
       [  0.        ,   0.        ,   0.        ,   0.        ,
          0.        ,   0.        ,   1.        ,   4.        ,
          0.        ,   0.        ],
       [  0.        ,   0.        ,   0.        ,   0.        ,
          0.        ,   0.        ,   0.        ,   0.        ,
          1.        ,   4.        ]])

R = array([0., 0., 0., 0., 0., 1., 1., 1., 1., 1.])

I try to solve this equation by using the following method:

import numpy as np
P = np.linalg.solve(W, R)

The result looks like:

P = array([ 1.29096548, -0.07274137, -1.82110745,  0.70527686, -1.80496726,
        0.70124182,  0.59473423,  0.10131644,  4.10879343, -0.77719836])

Problem:

However, the result P is not I expected.

As you see, here the elements of P denote the probability belonging to each class and they should vary from 0 to 1. I don't know how to add these constraints to my equation.

I find that the condition number of W is 537. Does it cause the problem?