Oh, How Long It Takes?

Algebra Level 5

x22212+y22232+z22252+w22272 = 1x24212+y24232+z24252+w24272 = 1x26212+y26232+z26252+w26272 = 1x28212+y28232+z28252+w28272 = 1\begin{aligned} \dfrac{x^2}{2^2-1^2}+\dfrac{y^2}{2^2-3^2}+\dfrac{z^2}{2^2-5^2}+\dfrac{w^2}{2^2-7^2} \ = \ 1 \\ \dfrac{x^2}{4^2-1^2}+\dfrac{y^2}{4^2-3^2}+\dfrac{z^2}{4^2-5^2}+\dfrac{w^2}{4^2-7^2} \ = \ 1 \\ \dfrac{x^2}{6^2-1^2}+\dfrac{y^2}{6^2-3^2}+\dfrac{z^2}{6^2-5^2}+\dfrac{w^2}{6^2-7^2} \ = \ 1 \\ \dfrac{x^2}{8^2-1^2}+\dfrac{y^2}{8^2-3^2}+\dfrac{z^2}{8^2-5^2}+\dfrac{w^2}{8^2-7^2} \ = \ 1 \\ \end{aligned}

Determine w2+x2+y2+z2w^2+x^2+y^2+z^2 if they satisfy the system of equations above.

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