Determine the system solution x+y+z=1 ax+by+cz=1 ax^2+b^2y+c^2z=1

If we create the matrix of the system and after that we
calculate it's determinant,we'll     note that the determinant is Vandermonde type, that
means that the solution is the result of multiplication of the 3 factors:
(b-a)(c-a)(c-b) and this one is different from zero, if a different from b and different
from c.

In this case , the system has just a single
solution and it's determined. We could solve the system using the Cramer method,
where:

x = det 1/det A

Y = det
2 / det A

z = det 3 / det
A

det1, det2, det3 - determinants formed from the initial
matrix of a system, where the column formed by the coefficients of the searched unknown
is substituted by the column of the free terms.

det A -
determinant of system matrix.

det1, det2, det3 - also
Vandermonde determinants.

det1 = (b-1)(c-1)(c-b) (we've
substituted the column formed by the coefficients 1, a and a^2 with the column of free
terms: 1,1,1)

det2 =
(1-a)(c-a)(c-1)

det3 =
(1-a)(1-b)(b-a)

x=(b-1)(c-1)(c-b) / (b-a)(c-a)(c-b) =
(b-1)(c-1) / (b-a)(c-a)

y = (1-a)(c-a)(c-1) /
(b-a)(c-a)(c-b)

y =(1-a)(c-1)
/(b-a)(c-b)

z = (1-a)(1-b)(b-a) /
(b-a)(c-a)(c-b)

z = (1-a)(1-b) /
(c-a)(c-b)