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

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

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

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

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

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

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

Multiply ${\displaystyle ab}$ by ${\displaystyle c}$.

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

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle ac+ab=bc}$.

Factor ${\displaystyle a}$ out of ${\displaystyle ac+ab}$.

Divide each term in ${\displaystyle a(c+b)=bc}$ by ${\displaystyle c+b}$ and simplify.