Solve by Isolating the Absolute Value for c a=b(1/c-1/d)

Apply the distributive property.

Combine ${\displaystyle b}$ and ${\displaystyle \frac{1}{c}}$.

Rewrite using the commutative property of multiplication.

Combine ${\displaystyle \frac{1}{d}}$ and ${\displaystyle b}$.

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

Since ${\displaystyle c,1,d}$ 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 ${\displaystyle c^1,d^1}$.

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

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

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

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

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

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle acd+bc=bd}$.

Factor ${\displaystyle c}$ out of ${\displaystyle acd+bc}$.

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