Eq. \ref{eq:5427510} is an initial equation. equation 1-13 on page 21 in \cite{1999_Tipler_Llewellyn} $$x' = \gamma (x - v t) \label{eq:5427510}$$ Eq. \ref{eq:2283140} is an initial equation. equation 1-14 on page 21 in \cite{1999_Tipler_Llewellyn} $$x = \gamma (x' + v t') \label{eq:2283140}$$ Substitute LHS of Eq. \ref{eq:5427510} into Eq. \ref{eq:2283140}; yields Eq. \ref{eq:4471422}. solve output expr for t' $$x = \gamma ( \gamma (x - v t) + v t' ) \label{eq:4471422}$$ Simplify Eq. \ref{eq:4471422}; yields Eq. \ref{eq:7169020}. $$x = \gamma ( \gamma x - \gamma v t + v t' ) \label{eq:7169020}$$ Simplify Eq. \ref{eq:7169020}; yields Eq. \ref{eq:6463955}. $$x = \gamma^2 x - \gamma^2 v t + \gamma v t' \label{eq:6463955}$$ Subtract $$\gamma^2 x$$ from both sides of Eq. \ref{eq:6463955}; yields Eq. \ref{eq:8494407}. $$x - \gamma^2 x = - \gamma^2 v t + \gamma v t' \label{eq:8494407}$$ Factor $$x$$ from the LHS of Eq. \ref{eq:8494407}; yields Eq. \ref{eq:3992172}. $$x (1 - \gamma^2 ) = - \gamma^2 v t + \gamma v t' \label{eq:3992172}$$ Add $$\gamma^2 v t$$ to both sides of Eq. \ref{eq:3992172}; yields Eq. \ref{eq:6047713}. $$x (1 - \gamma^2 ) + \gamma^2 v t = \gamma v t' \label{eq:6047713}$$ Divide both sides of Eq. \ref{eq:6047713} by $$\gamma v$$; yields Eq. \ref{eq:5995189}. $$\frac{x (1 - \gamma^2 )}{\gamma v} + \frac{\gamma^2 v t}{\gamma v} = t' \label{eq:5995189}$$ Simplify Eq. \ref{eq:5995189}; yields Eq. \ref{eq:7546640}. $$\frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t = t' \label{eq:7546640}$$ Swap LHS of Eq. \ref{eq:7546640} with RHS; yields Eq. \ref{eq:1693888}. $$t' = \frac{\gamma x (1 - \gamma^2 )}{\gamma^2 v} + \gamma t \label{eq:1693888}$$ Eq. \ref{eq:6319661} is an initial equation. $$x^2 + y^2 + z^2 = c^2 t^2 \label{eq:6319661}$$ Eq. \ref{eq:5649086} is an initial equation. $$x'^2 + y'^2 + z'^2 = c^2 t'^2 \label{eq:5649086}$$ Eq. \ref{eq:6316097} is an assumption. $$y' = y \label{eq:6316097}$$ Eq. \ref{eq:7666907} is an assumption. $$z' = z \label{eq:7666907}$$ Substitute LHS of Eq. \ref{eq:7666907} and LHS of Eq. \ref{eq:6316097} and LHS of Eq. \ref{eq:1693888} and LHS of Eq. \ref{eq:5427510} into Eq. \ref{eq:5649086}; yields Eq. \ref{eq:4326342}. $$\gamma^2 (x - v t)^2 + y^2 + z^2 = c^2 \gamma^2 \left( t + \frac{ 1 - \gamma^2 }{ \gamma^2 } \frac{x}{v} \right)^2 \label{eq:4326342}$$ Simplify Eq. \ref{eq:4326342}; yields Eq. \ref{eq:6066191}. expanded the squared terms $$\gamma^2 x^2 - \gamma^2 2 x v t + \gamma^2 v^2 t^2 + y^2 + z^2 = c^2 \gamma^2 \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x^2}{\gamma^2} + c^2 \gamma^2 2 t \left(\frac{1-\gamma^2}{\gamma^2}\right)\frac{x}{\gamma} + c^2 \gamma^2 t^2 \label{eq:6066191}$$ Simplify Eq. \ref{eq:6066191}; yields Eq. \ref{eq:4202425}. grouped by terms for x^2, xt, and t^2 $$\left( \gamma^2 - c^2 \gamma^2 \left( \frac{1-\gamma^2}{\gamma^2} \right)^2 \frac{1}{v^2} \right) x^2 + y^2 + z^2 + \left( -\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} \right) = t^2 \left( c^2 \gamma^2 - \gamma^2 v^2 \right) \label{eq:4202425}$$ Eq. \ref{eq:6319661} is equivalent to Eq. \ref{eq:4202425} under the condition in Eq. \ref{eq:2562123}. based on the comparison of the x^2 terms $$\gamma^2 - c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 \label{eq:2562123}$$ Eq. \ref{eq:6319661} is equivalent to Eq. \ref{eq:4202425} under the condition in Eq. \ref{eq:3640931}. based on the comparison of the t^2 terms $$-\gamma^2 v^2 + c^2 \gamma^2 = c^2 \label{eq:3640931}$$ Eq. \ref{eq:6319661} is equivalent to Eq. \ref{eq:4202425} under the condition in Eq. \ref{eq:6685577}. based on the comparison of the (x t) terms $$-\gamma^2 2 x v t - c^2 \gamma^2 2 t \left( \frac{1-\gamma^2}{\gamma^2} \right) \frac{x}{v} = 0 \label{eq:6685577}$$ Subtract $$\gamma^2$$ from both sides of Eq. \ref{eq:2562123}; yields Eq. \ref{eq:7403799}. solve for \gamma $$- c^2 \gamma^2 \frac{(1-\gamma^2)^2}{v^2 \gamma^4} = 1 - \gamma^2 \label{eq:7403799}$$ Divide both sides of Eq. \ref{eq:7403799} by $$1 - \gamma^2$$; yields Eq. \ref{eq:4052253}. $$- c^2 \frac{(1-\gamma^2)}{v^2 \gamma^2} = 1 \label{eq:4052253}$$ Multiply both sides of Eq. \ref{eq:4052253} by $$v^2 \gamma^2$$; yields Eq. \ref{eq:8195408}. $$-c^2 + c^2 \gamma^2 = v^2 \gamma^2 \label{eq:8195408}$$ Add $$c^2$$ to both sides of Eq. \ref{eq:8195408}; yields Eq. \ref{eq:6913493}. $$c^2 \gamma^2 = v^2 \gamma^2 + c^2 \label{eq:6913493}$$ Subtract $$v^2 \gamma^2$$ from both sides of Eq. \ref{eq:6913493}; yields Eq. \ref{eq:5207615}. $$c^2 \gamma^2 - v^2 \gamma^2 = c^2 \label{eq:5207615}$$ Factor $$\gamma^2$$ from the LHS of Eq. \ref{eq:5207615}; yields Eq. \ref{eq:8842089}. $$\gamma^2 (c^2 - v^2) = c^2 \label{eq:8842089}$$ Divide both sides of Eq. \ref{eq:8842089} by $$c^2 - \gamma^2$$; yields Eq. \ref{eq:7595841}. $$\gamma^2 = \frac{c^2}{c^2 - \gamma^2} \label{eq:7595841}$$ Take the square root of both sides of Eq. \ref{eq:7595841}; yields Eq. \ref{eq:3040283} and Eq. \ref{eq:6010461}. $$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} \label{eq:3040283}$$ $$\gamma = \frac{-1}{\sqrt{1-\frac{v^2}{c^2}}} \label{eq:6010461}$$ Eq. \ref{eq:3040283} is one of the final equations. Lorentz factor definition