Solve the equations
$x + a y + a^{2} z = a^{3}, x + b y + b^{2} z = b^{3}, x + c y + c^{2} z = c^{3}$ .

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Solution

$Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{2} 1 & b & b^{2} 1 & c & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a)$
$Δ_{1} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} a^{3} & a & a^{2} b^{3} & b & b^{2} c^{3} & c & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = a b c (a - b) (b - c) (c - a)$
$Δ_{2} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a^{3} & a^{2} 1 & b^{3} & b^{2} 1 & c^{3} & c^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = - (a - b) (b - c) (c - a) (a b + b c + c a)$

$Δ_{3} = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & a & a^{3} 1 & b & b^{3} 1 & c & c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣ = (a - b) (b - c) (c - a) (a + b + c)$
$x = \frac{Δ_{1}}{Δ} = a b c; y = \frac{Δ_{2}}{Δ} = - (a b + b c + c a); z = \frac{Δ_{3}}{Δ} = (a + b + c)$

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a, b, c

x, y, z

x + a y + a^{2} z = a^{3}

x + b y + b^{2} z = b^{3}

x + c y + c^{2} z = c^{3}

z + a y = a^{2} x + a^{3} = 0, z + b y + b^{2} x + b^{3} = 0, z + c y + c^{2} x + c^{3} = 0.

z + a y + a^{2} x + a^{3} = 0 z + b y + b^{2} x + b^{3} = 0 z + c y + c^{2} x + c^{3} = 0

(a^{3} + b^{3} + c^{3} - 3 a b c) (x^{3} + y^{3} + z^{3} - 3 x y z) = (X^{3} + Y^{3} + Z^{3} - 3 X Y Z)

(x - a)^{3} + (x - b)^{3} + (x - c)^{3} = 0

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