System of equations

Algebra Level 4

$\begin{cases} {y = \sqrt { x+1 } +\sqrt [ 4 ]{ x-1 } -\sqrt { { y }^{ 4 }+2 } } \\ {0=x^2+2x(y-1)+y^2-6y+1} \end{cases}$

Let $(x_1,y_1), (x_2,y_2), \ldots , (x_n,y_n)$ be the solution set of the above system of equations for real $x,y$

Evaluate $\displaystyle n+ \sum _{ i=1 }^{ n }{ (x_i+y_i) }$

The answer is 6.0.

1 solution

Nguyen Thanh Long
Feb 18, 2015

According to second equation, having: $x=2\sqrt{y}-(y-1)$ Replacing to first equation: $\sqrt{-y+2\sqrt{y}+2}+(-y+2\sqrt{y})^{\frac{1}{4}}=\sqrt{y^4+2}+y$ $\Rightarrow 0 \le y \le 1$ It is easy to get two roots of (x, y): (1, 0), (2, 1) $n+\Sigma^{n}_{i=1} (x_{i} + y_{i})=\boxed{6}$

System of equations

The answer is 6.0.

1 solution

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