Solve for sᵢ 1/f=1/s_o+1/s_i

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

Since ${\displaystyle s_i,f,s_o}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 1,1,1}$ then find LCM for the variable part ${\displaystyle {s_i}^1,f^1,{s_o}^1}$.

The LCM is the smallest positive number that all of the numbers divide into evenly.

The number ${\displaystyle 1}$ is not a prime number because it only has one positive factor, which is itself.

The LCM of ${\displaystyle 1,1,1}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factor for ${\displaystyle {s_i}^1}$ is ${\displaystyle s_i}$ itself.

The factor for ${\displaystyle f^1}$ is ${\displaystyle f}$ itself.

The factor for ${\displaystyle {s_o}^1}$ is ${\displaystyle s_o}$ itself.

The LCM of ${\displaystyle {s_i}^1,f^1,{s_o}^1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

Multiply ${\displaystyle s_{i}f}$ by ${\displaystyle s_o}$.

Multiply each term in ${\displaystyle \frac{1}{s_i}=\frac{1}{f}-\frac{1}{s_o}}$ by ${\displaystyle s_{i}f{s}_o}$.

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle s_i{s}_o-s_{i}f=f{s}_o}$.

Factor ${\displaystyle s_i}$ out of ${\displaystyle s_i{s}_o-s_{i}f}$.

Rewrite ${\displaystyle -1f}$ as ${\displaystyle -f}$.

Divide each term in ${\displaystyle s_i(s_o-f)=f{s}_o}$ by ${\displaystyle s_o-f}$ and simplify.