Solve for p 1/p+1/q=1/f

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

Since ${\displaystyle p,f,q}$ 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 ${\displaystyle p^1,f^1,q^1}$.

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 ${\displaystyle p^1}$ is ${\displaystyle p}$ itself.

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

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

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

Multiply ${\displaystyle pf}$ by ${\displaystyle q}$.

Multiply each term in ${\displaystyle \frac{1}{p}=\frac{1}{f}-\frac{1}{q}}$ by ${\displaystyle pfq}$.

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle pq-pf=fq}$.

Factor ${\displaystyle p}$ out of ${\displaystyle pq-pf}$.

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

Divide each term in ${\displaystyle p(q-f)=fq}$ by ${\displaystyle q-f}$ and simplify.