$x + y + z = 1$
$a x + b y + c z = k$
$a^{2} x + b^{2} y + c^{2} z = k^{2}$
Solve by Crammer's rule

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Solution

The given equations are

$x + y + z = 1$

$a x + b y + c z = k$

$a^{2} x + b^{2} y + c^{2} z = k^{2}$

or

$A X = B$

By cramer's rule

$x = \frac{Δ_{x}}{Δ}$ $y = \frac{Δ_{y}}{Δ}$ $z = \frac{Δ_{z}}{Δ}$

By solving the matrices, we get,

$| A | = (a - b) (b - c) (c - a)$

$∴ x = \frac{Δ_{x}}{| A |} = \frac{(k - b) (c - k}{(a - b) (c - a)}$

$y = \frac{Δ_{y}}{| A |} = \frac{(a - k) (k - c)}{(a - b) (b - c)}$

$z = \frac{Δ_{z}}{| A |} = \frac{(b - k) (k - a)}{(b - c) (c - a)}$

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a x + b y + c z = k

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

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

x + y + z = 1

a x + b y + c z = k

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

x + y + z + u = 1

a x + b y + c z + d u = k

a^{2} x + b^{2} y + c^{2} z + d^{2} u = k^{2}

a^{3} x + b^{3} y + c^{3} z + d^{3} u = k^{3}

a x + b y + c z = k

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

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

a x + b y + c z = a^{2} x + b^{2} y + c^{z} = 0, x + y + z + (b - c) (c - a) (a - b) = 0 .

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