Solve for b 1/a+1/b=c for b

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

Since ${\displaystyle b,1,a}$ 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 ${\displaystyle b^1,a^1}$.

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

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

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

Multiply ${\displaystyle b}$ by ${\displaystyle a}$.

Multiply each term in ${\displaystyle \frac{1}{b}=c-\frac{1}{a}}$ by ${\displaystyle ba}$.

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle cba-b=a}$.

Factor ${\displaystyle b}$ out of ${\displaystyle cba-b}$.

Divide each term in ${\displaystyle b(ca-1)=a}$ by ${\displaystyle ca-1}$ and simplify.