Is it possible to solve this system of equations in terms of p?

Question

Given $$ \begin{aligned} b(1-p) +cp &= a \\ ap +c(1-p) &= b \\ a(1-p) + bp &= c \\ a+b+c&=1 \end{aligned} $$

Is it possible to solve a, b, and c, individually, only in terms of p?

Answer 1

1 = a + b + c

1 = [b - bp + cp] + [c - cp + ap] + [a - ap + bp]

Substituting the p terms for a, b, and c, we find the p terms cancel.

p can be any number. It does not affect the value of a, b, or c.

If p = 0, then a = b = c