Solve L_{f}=L_{0}(1+alphaDeltat)

$$L _ { f } = L _ { 0 } ( 1 + \alpha \Delta t )$$

Solve for L_0

$$\left\{\begin{matrix}L_{0}=\frac{L_{f}}{t\Delta \alpha +1}\text{, }&t=0\text{ or }\Delta =0\text{ or }\alpha \neq -\frac{1}{t\Delta }\\L_{0}\in \mathrm{R}\text{, }&L_{f}=0\text{ and }\alpha =-\frac{1}{t\Delta }\text{ and }t\neq 0\text{ and }\Delta \neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply L_{0} by 1+\alpha \Delta t.

$$L_{f}=L_{0}+L_{0}\alpha \Delta t$$

Swap sides so that all variable terms are on the left hand side.

$$L_{0}+L_{0}\alpha \Delta t=L_{f}$$

Combine all terms containing L_{0}.

$$\left(1+\alpha \Delta t\right)L_{0}=L_{f}$$

The equation is in standard form.

$$\left(t\Delta \alpha +1\right)L_{0}=L_{f}$$

Divide both sides by 1+\alpha \Delta t.

$$\frac{\left(t\Delta \alpha +1\right)L_{0}}{t\Delta \alpha +1}=\frac{L_{f}}{t\Delta \alpha +1}$$

Dividing by 1+\alpha \Delta t undoes the multiplication by 1+\alpha \Delta t.

$$L_{0}=\frac{L_{f}}{t\Delta \alpha +1}$$

Solve for L_f

$$L_{f}=L_{0}\left(t\Delta \alpha +1\right)$$