Solve P_{50}=L+[frac{1/100-c_{F}b}{f}]i

$$P _ { 50 } = L + [ \frac { \frac { 1 } { 100 } - c _ { F } b } { f } ] i$$

Solve for L

$$L=\frac{ibc_{F}}{f}+P_{50}+\frac{-i}{100f}$$

$$f\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by f.

$$P_{50}f=fL+\left(\frac{1}{100}-c_{F}b\right)i$$

Use the distributive property to multiply \frac{1}{100}-c_{F}b by i.

$$P_{50}f=fL+\frac{1}{100}i-ibc_{F}$$

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

$$fL+\frac{1}{100}i-ibc_{F}=P_{50}f$$

Subtract \frac{1}{100}i from both sides.

$$fL-ibc_{F}=P_{50}f-\frac{1}{100}i$$

Subtract -ibc_{F} from both sides.

$$fL=P_{50}f-\frac{1}{100}i-\left(-ibc_{F}\right)$$

Multiply -1 and -i to get i.

$$fL=P_{50}f-\frac{1}{100}i+ibc_{F}$$

The equation is in standard form.

$$fL=P_{50}f+ibc_{F}-\frac{1}{100}i$$

Divide both sides by f.

$$\frac{fL}{f}=\frac{P_{50}f+ibc_{F}-\frac{1}{100}i}{f}$$

Dividing by f undoes the multiplication by f.

$$L=\frac{P_{50}f+ibc_{F}-\frac{1}{100}i}{f}$$

Divide P_{50}f-\frac{1}{100}i+ibc_{F} by f.

$$L=\frac{ibc_{F}-\frac{1}{100}i}{f}+P_{50}$$

Solve for P_50

$$P_{50}=-\frac{ibc_{F}}{f}+L+\frac{i}{100f}$$

$$f\neq 0$$