$$\frac{ 1 }{ 5 \sqrt{ 2 } -7 } =x \sqrt{ 2 } +y$$
$x=-\frac{\sqrt{2}\left(y-5\sqrt{2}-7\right)}{2}$
$$\frac{5\sqrt{2}+7}{\left(5\sqrt{2}-7\right)\left(5\sqrt{2}+7\right)}=x\sqrt{2}+y$$
$$\frac{5\sqrt{2}+7}{\left(5\sqrt{2}\right)^{2}-7^{2}}=x\sqrt{2}+y$$
$$\frac{5\sqrt{2}+7}{5^{2}\left(\sqrt{2}\right)^{2}-7^{2}}=x\sqrt{2}+y$$
$$\frac{5\sqrt{2}+7}{25\left(\sqrt{2}\right)^{2}-7^{2}}=x\sqrt{2}+y$$
$$\frac{5\sqrt{2}+7}{25\times 2-7^{2}}=x\sqrt{2}+y$$
$$\frac{5\sqrt{2}+7}{50-7^{2}}=x\sqrt{2}+y$$
$$\frac{5\sqrt{2}+7}{50-49}=x\sqrt{2}+y$$
$$\frac{5\sqrt{2}+7}{1}=x\sqrt{2}+y$$
$$5\sqrt{2}+7=x\sqrt{2}+y$$
$$x\sqrt{2}+y=5\sqrt{2}+7$$
$$x\sqrt{2}=5\sqrt{2}+7-y$$
$$\sqrt{2}x=-y+5\sqrt{2}+7$$
$$\frac{\sqrt{2}x}{\sqrt{2}}=\frac{-y+5\sqrt{2}+7}{\sqrt{2}}$$
$$x=\frac{-y+5\sqrt{2}+7}{\sqrt{2}}$$
$$x=\frac{\sqrt{2}\left(-y+5\sqrt{2}+7\right)}{2}$$
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$y=-\sqrt{2}x+5\sqrt{2}+7$
