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

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

Solve for L

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

$$f\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by f.

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

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

$$P_{50}f=fL+i\times \frac{1n}{100}-ic_{F}b$$

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

$$fL+i\times \frac{1n}{100}-ic_{F}b=P_{50}f$$

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

$$fL-ic_{F}b=P_{50}f-i\times \frac{1n}{100}$$

Subtract -ic_{F}b from both sides.

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

Multiply both sides of the equation by 100.

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

Reorder the terms.

$$100Lf=100\left(P_{50}f-i\times \frac{n}{100}\right)-\left(-100ibc_{F}\right)$$

Multiply both sides of the equation by 100.

$$10000Lf=10000\left(P_{50}f-i\times \frac{n}{100}\right)-100\times \left(-100i\right)bc_{F}$$

Use the distributive property to multiply 10000 by P_{50}f-i\times \frac{n}{100}.

$$10000Lf=10000P_{50}f-10000i\times \frac{n}{100}-100\times \left(-100i\right)bc_{F}$$

Multiply -100 and -100i to get 10000i.

$$10000Lf=10000P_{50}f-10000i\times \frac{n}{100}+10000ibc_{F}$$

Multiply both sides of the equation by 100.

$$1000000Lf=1000000P_{50}f-10000in+1000000ibc_{F}$$

The equation is in standard form.

$$1000000fL=1000000P_{50}f+1000000ibc_{F}-10000in$$

Divide both sides by 1000000f.

$$\frac{1000000fL}{1000000f}=\frac{1000000P_{50}f+1000000ibc_{F}-10000in}{1000000f}$$

Dividing by 1000000f undoes the multiplication by 1000000f.

$$L=\frac{1000000P_{50}f+1000000ibc_{F}-10000in}{1000000f}$$

Divide 1000000P_{50}f-10000in+1000000ic_{F}b by 1000000f.

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

Solve for P_50

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

$$f\neq 0$$