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

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

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

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

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

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

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

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

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

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle rb+ra=ab}$.

Factor ${\displaystyle r}$ out of ${\displaystyle rb+ra}$.

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