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

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

Solve for L

$$L=\frac{ibc_{P}}{f}+P_{50}-\frac{im}{100f}$$

$$f\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by f.

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

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

$$P_{50}f=fL+i\times \frac{m}{100}-ic_{P}b$$

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

$$fL+i\times \frac{m}{100}-ic_{P}b=P_{50}f$$

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

$$fL-ic_{P}b=P_{50}f-i\times \frac{m}{100}$$

Subtract -ic_{P}b from both sides.

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

Multiply both sides of the equation by 100.

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

Multiply both sides of the equation by 100.

$$10000fL=10000\left(P_{50}f-i\times \frac{m}{100}\right)-100\times \left(-100i\right)c_{P}b$$

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

$$10000fL=10000P_{50}f-10000i\times \frac{m}{100}-100\times \left(-100i\right)c_{P}b$$

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

$$10000fL=10000P_{50}f-10000i\times \frac{m}{100}+10000ic_{P}b$$

Multiply both sides of the equation by 100.

$$1000000fL=1000000P_{50}f-10000im+1000000ic_{P}b$$

The equation is in standard form.

$$1000000fL=1000000P_{50}f+1000000ibc_{P}-10000im$$

Divide both sides by 1000000f.

$$\frac{1000000fL}{1000000f}=\frac{1000000P_{50}f+1000000ibc_{P}-10000im}{1000000f}$$

Dividing by 1000000f undoes the multiplication by 1000000f.

$$L=\frac{1000000P_{50}f+1000000ibc_{P}-10000im}{1000000f}$$

Divide 1000000P_{50}f-10000im+1000000ic_{P}b by 1000000f.

$$L=\frac{ibc_{P}-\frac{im}{100}}{f}+P_{50}$$

Solve for P_50

$$P_{50}=-\frac{ibc_{P}}{f}+L+\frac{im}{100f}$$

$$f\neq 0$$