\(\dfrac{\sqrt{2}+7}{9}+2\left(\sqrt{2}+1\right)y-\sqrt{2}-1=0\)
\(\Leftrightarrow2\left(\sqrt{2}+1\right)y=\sqrt{2}+1-\dfrac{\sqrt{2}+7}{9}\)
\(\Leftrightarrow2\left(\sqrt{2}+1\right)y=\dfrac{9\left(\sqrt{2}+1\right)-\sqrt{2}-7}{9}\)
\(\Leftrightarrow2\left(\sqrt{2}+1\right)y=\dfrac{9\sqrt{2}+9-\sqrt{2}-7}{9}\)
\(\Leftrightarrow2\left(\sqrt{2}+1\right)y=\dfrac{8\sqrt{2}+2}{9}\)
\(\Leftrightarrow y=\dfrac{8\sqrt{2}+2}{9}\cdot\dfrac{1}{2\left(\sqrt{2}+1\right)}\)
\(\Leftrightarrow y=\dfrac{2\left(4\sqrt{2}+1\right)}{9}\cdot\dfrac{1}{2\left(\sqrt{2}+1\right)}\)
\(\Leftrightarrow y=\dfrac{4\sqrt{2}+1}{9\left(\sqrt{2}+1\right)}\)
\(\Leftrightarrow y=\dfrac{\left(4\sqrt{2}+1\right)\left(\sqrt{2}-1\right)}{9\cdot\left(2-1\right)}\)
\(\Leftrightarrow y=\dfrac{8-4\sqrt{2}+\sqrt{2}-1}{9}\)
\(\Leftrightarrow y=\dfrac{7-3\sqrt{2}}{9}\)
\(\Leftrightarrow2\left(\sqrt{2}+1\right)y=\sqrt{2}+1-\dfrac{\sqrt{2}+7}{9}=\dfrac{2+8\sqrt{2}}{9}\)
\(\Rightarrow y=\dfrac{3\sqrt{2}-7}{9}\)
