Solve 1/L=1/L_{1}+1/2L_{2}

$$\frac { 1 } { L } = \frac { 1 } { L _ { 1 } } + \frac { 1 } { 2 L _ { 2 } }$$

Solve for L

$$L=\frac{2L_{1}L_{2}}{L_{1}+2L_{2}}$$

$$L_{2}\neq 0\text{ and }L_{1}\neq 0\text{ and }L_{1}\neq -2L_{2}$$

Steps for Solving Linear Equation

Variable L cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2LL_{1}L_{2}, the least common multiple of L,L_{1},2L_{2}.

$$2L_{1}L_{2}=2LL_{2}+LL_{1}$$

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

$$2LL_{2}+LL_{1}=2L_{1}L_{2}$$

Combine all terms containing L.

$$\left(2L_{2}+L_{1}\right)L=2L_{1}L_{2}$$

The equation is in standard form.

$$\left(L_{1}+2L_{2}\right)L=2L_{1}L_{2}$$

Divide both sides by L_{1}+2L_{2}.

$$\frac{\left(L_{1}+2L_{2}\right)L}{L_{1}+2L_{2}}=\frac{2L_{1}L_{2}}{L_{1}+2L_{2}}$$

Dividing by L_{1}+2L_{2} undoes the multiplication by L_{1}+2L_{2}.

$$L=\frac{2L_{1}L_{2}}{L_{1}+2L_{2}}$$

Variable L cannot be equal to 0.

$$L=\frac{2L_{1}L_{2}}{L_{1}+2L_{2}}\text{, }L\neq 0$$

Solve for L_1

$$L_{1}=-\frac{2LL_{2}}{L-2L_{2}}$$

$$L_{2}\neq 0\text{ and }L\neq 0\text{ and }L\neq 2L_{2}$$

Steps for Solving Linear Equation

Variable L_{1} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2LL_{1}L_{2}, the least common multiple of L,L_{1},2L_{2}.

$$2L_{1}L_{2}=2LL_{2}+LL_{1}$$

Subtract LL_{1} from both sides.

$$2L_{1}L_{2}-LL_{1}=2LL_{2}$$

Combine all terms containing L_{1}.

$$\left(2L_{2}-L\right)L_{1}=2LL_{2}$$

Divide both sides by 2L_{2}-L.

$$\frac{\left(2L_{2}-L\right)L_{1}}{2L_{2}-L}=\frac{2LL_{2}}{2L_{2}-L}$$

Dividing by 2L_{2}-L undoes the multiplication by 2L_{2}-L.

$$L_{1}=\frac{2LL_{2}}{2L_{2}-L}$$

Variable L_{1} cannot be equal to 0.

$$L_{1}=\frac{2LL_{2}}{2L_{2}-L}\text{, }L_{1}\neq 0$$