Question

Solve the equations:
$x+y+z+u=1$,
$ax+by+cz+du=k$,
$a^{2}x+b^{2}y+c^{2}z+d^{2}u=k^2$,
$a^{3}x+b^{3}y+c^{3}z+d^{3}u=k^3$.

Answer 1

$\Delta =\left|\begin{array}{cccc}1& 1& 1& 1\\ a& b& c& d\\ a^2& b^2& c^2& d^2\\ a^3& b^3& c^3& d^3\end{array}\right|=(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$

${\Delta }_1=\left|\begin{array}{cccc}1& 1& 1& 1\\ k& b& c& d\\ k^2& b^2& c^2& d^2\\ k^3& b^3& c^3& d^3\end{array}\right|=-(b-k)(c-k)(d-k)(b-c)(b-d)(c-d)$

${\Delta }_2=\left|\begin{array}{cccc}1& 1& 1& 1\\ a& k& c& d\\ a^2& k^2& c^2& d^2\\ a^3& k^3& c^3& d^3\end{array}\right|=-(a-k)(c-k)(d-k)(a-c)(a-d)(c-d)$

${\Delta }_3=\left|\begin{array}{cccc}1& 1& 1& 1\\ a& b& k& d\\ a^2& b^2& k^2& d^2\\ a^3& b^3& k^3& d^3\end{array}\right|=-(a-k)(b-k)(d-k)(a-b)(a-d)(b-d)$

${\Delta }_4=\left|\begin{array}{cccc}1& 1& 1& 1\\ a& b& c& k\\ a^2& b^2& c^2& k^2\\ a^3& b^3& c^3& k^3\end{array}\right|=-(a-k)(b-k)(c-k)(a-b)(a-c)(b-c)$

$x=\frac{{\Delta }_1}{\Delta }=-\frac{(b-k)(c-k)(d-k)}{a-b\left)\right(a-c\left)\right(a-d)};y=\frac{{\Delta }_2}{\Delta }=-\frac{(a-k)(c-k)(d-k)}{a-b\left)\right(b-c\left)\right(b-d)};z=\frac{{\Delta }_3}{\Delta }=-\frac{(a-k)(b-k)(d-k)}{a-c\left)\right(b-c\left)\right(c-d)};u=\frac{{\Delta }_4}{\Delta }=-\frac{(a-k)(b-k)(c-k)}{a-d\left)\right(b-d\left)\right(c-d)}$