Example Question - variable substitution

Algebraic Expression Simplification

\[ \begin{array}{l} \text{Given: } y + \frac{1}{y} = b \\ \text{Squaring both sides: } \\ \left( y + \frac{1}{y} \right)^2 = b^2 \\ y^2 + 2 \cdot y \cdot \frac{1}{y} + \left( \frac{1}{y} \right)^2 = b^2 \\ y^2 + 2 + \frac{1}{y^2} = b^2 \\ \text{Subtract 2 from both sides: } \\ y^2 + \frac{1}{y^2} = b^2 - 2 \end{array} \]