Solve for a 2/a+1/b=13/15

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

Since ${\displaystyle a,15,b}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 1,15,1}$ then find LCM for the variable part ${\displaystyle a^1,b^1}$.

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.

${\displaystyle 15}$ has factors of ${\displaystyle 3}$ and ${\displaystyle 5}$.

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

The LCM of ${\displaystyle 1,15,1}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

Multiply ${\displaystyle 3}$ by ${\displaystyle 5}$.

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

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

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

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

The LCM for ${\displaystyle a,15,b}$ is the numeric part ${\displaystyle 15}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{2}{a}=\frac{13}{15}-\frac{1}{b}}$ by ${\displaystyle 15ab}$.

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle 13ab-15a=30b}$.

Factor ${\displaystyle a}$ out of ${\displaystyle 13ab-15a}$.

Divide each term in ${\displaystyle a(13b-15)=30b}$ by ${\displaystyle 13b-15}$ and simplify.