Solve the system of the equations: `a x+b y+c z=d` `a^2x+b^2y+c^2z=d^2` `a^3x+b^3y+c^3z=d^3` Will the solution always exist and be unique?

Explanation

Here we use cramer's rule to solve this system
`Delta = |{:(a,, b,,c),(a^(2),,b^(2),,c^(2)),(a^(3),,b^(3),,c^(3)):}|`
`= abc |{:(1,, 1,,1),(a,,b,,c),(a^(2),,b^(2),,c^(2)):}| `
`= abc (a-c)(b-c)(c-a)`
If a,b,c are distinct and nonzero then the system has a unique solution Now ,
`Delta_(x)= |{:(d,,b,,c),(d^(2),,b^(2),,c^(2)),(d^(3),,b^(3),,c^(3)):}|= dbc (d-b)(b-c)(c-d)`
`:. x=(Delta_(x))/(Delta) =(d(d-b)(c-d))/(a(a-b)(c-d))`
By symmetry we have
`y=(Delta_(y))/(Delta)=(d(a-d)(d-c))/(b(a-b)(b-c))`
`z=(Delta_(z))/(Delta)=(d(b-d)(d-a))/(c(b-c)(c-a))`