Eq. \ref{eq:7864125} is an initial equation. $$a = \frac{v - v_0}{t} \label{eq:7864125}$$ Multiply both sides of Eq. \ref{eq:7864125} by $$t$$; yields Eq. \ref{eq:5666935}. $$a t = v - v_0 \label{eq:5666935}$$ Add $$v_0$$ to both sides of Eq. \ref{eq:5666935}; yields Eq. \ref{eq:3386860}. $$a t + v_0 = v \label{eq:3386860}$$ Swap LHS of Eq. \ref{eq:3386860} with RHS; yields Eq. \ref{eq:8873965}. $$v = v_0 + a t \label{eq:8873965}$$ Eq. \ref{eq:8873965} is one of the final equations. Eq. \ref{eq:8658331} is an initial equation. $$v_{\rm average} = \frac{d}{t} \label{eq:8658331}$$ Eq. \ref{eq:5013638} is an initial equation. $$v_{\rm average} = \frac{v + v_0}{2} \label{eq:5013638}$$ LHS of Eq. \ref{eq:8658331} is equal to LHS of Eq. \ref{eq:5013638}; yields Eq. \ref{eq:4622149}. $$\frac{d}{t} = \frac{v + v_0}{2} \label{eq:4622149}$$ Multiply both sides of Eq. \ref{eq:4622149} by $$t$$; yields Eq. \ref{eq:1476448}. $$d = \left(\frac{v + v_0}{2}\right)t \label{eq:1476448}$$ Substitute RHS of Eq. \ref{eq:8873965} into Eq. \ref{eq:1476448}; yields Eq. \ref{eq:3069767}. $$d = \frac{(v_0 + a t) + v_0}{2} t \label{eq:3069767}$$ Simplify Eq. \ref{eq:3069767}; yields Eq. \ref{eq:6881977}. $$d = \frac{2 v_0 + a t}{2} t \label{eq:6881977}$$ Simplify Eq. \ref{eq:6881977}; yields Eq. \ref{eq:5385244}. $$d = v_0 t + \frac{1}{2} a t^2 \label{eq:5385244}$$ Eq. \ref{eq:5385244} is one of the final equations. Raise both sides of Eq. \ref{eq:8873965} to $$2$$; yields Eq. \ref{eq:4385757}. $$v^2 = v_0^2 + 2 a t v_0 + a^2 t^2 \label{eq:4385757}$$ Simplify Eq. \ref{eq:4385757}; yields Eq. \ref{eq:9796063}. factored 2a out of two terms $$v^2 = v_0^2 + 2 a \left( v_0 t +\frac{1}{2} a t^2 \right) \label{eq:9796063}$$ Substitute RHS of Eq. \ref{eq:9796063} into Eq. \ref{eq:5385244}; yields Eq. \ref{eq:7702534}. $$v^2 = v_0^2 + 2 a d \label{eq:7702534}$$ Eq. \ref{eq:7702534} is one of the final equations. Subtract $$v_0$$ from both sides of Eq. \ref{eq:8873965}; yields Eq. \ref{eq:4127918}. $$v - v_0 = a t \label{eq:4127918}$$ Swap LHS of Eq. \ref{eq:4127918} with RHS; yields Eq. \ref{eq:5666935}. $$a t = v - v_0 \label{eq:5666935}$$ Divide both sides of Eq. \ref{eq:5666935} by $$a$$; yields Eq. \ref{eq:8222540}. $$t = \frac{v - v_0}{a} \label{eq:8222540}$$ Substitute RHS of Eq. \ref{eq:8222540} into Eq. \ref{eq:1476448}; yields Eq. \ref{eq:9270356}. $$d = \frac{1}{2} (v + v_0) \left( \frac{v - v_0}{a} \right) \label{eq:9270356}$$ Simplify Eq. \ref{eq:9270356}; yields Eq. \ref{eq:7103968}. difference of squares $$d = \frac{1}{2 a} (v^2 - v_0^2) \label{eq:7103968}$$ Multiply both sides of Eq. \ref{eq:7103968} by $$2 a$$; yields Eq. \ref{eq:6814979}. $$2 a d = v^2 - v_0^2 \label{eq:6814979}$$ Add $$v_0^2$$ to both sides of Eq. \ref{eq:6814979}; yields Eq. \ref{eq:7086842}. $$2 a d + v_0^2 = v^2 \label{eq:7086842}$$ Swap LHS of Eq. \ref{eq:7086842} with RHS; yields Eq. \ref{eq:7702534}. $$v^2 = v_0^2 + 2 a d \label{eq:7702534}$$ Eq. \ref{eq:1476448} is one of the final equations. Subtract $$a t$$ from both sides of Eq. \ref{eq:8873965}; yields Eq. \ref{eq:8007427}. $$v - a t = v_0 \label{eq:8007427}$$ Substitute RHS of Eq. \ref{eq:8007427} into Eq. \ref{eq:5385244}; yields Eq. \ref{eq:5577530}. $$d = (v - a t) t + \frac{1}{2} a t^2 \label{eq:5577530}$$ Simplify Eq. \ref{eq:5577530}; yields Eq. \ref{eq:8442394}. $$d = v t - a t^2 + \frac{1}{2} a t^2 \label{eq:8442394}$$ Simplify Eq. \ref{eq:8442394}; yields Eq. \ref{eq:3917794}. $$d = v t - \frac{1}{2} a t^2 \label{eq:3917794}$$ Eq. \ref{eq:3917794} is one of the final equations.