Solve for y 1/(2y)-1/(6y^2)=-7/(6y^2)

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

Since ${\displaystyle 2y,6y^2,6y^2}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 2,6,6}$ then find LCM for the variable part ${\displaystyle y^1,y^2,y^2}$.

The LCM is the smallest positive number that all of the numbers divide into evenly.

Since ${\displaystyle 2}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 2}$.

${\displaystyle 6}$ has factors of ${\displaystyle 2}$ and ${\displaystyle 3}$.

Multiply ${\displaystyle 2}$ by ${\displaystyle 3}$.

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

The factors for ${\displaystyle y^2}$ are ${\displaystyle y\cdot y}$, which is ${\displaystyle y}$ multiplied by each other ${\displaystyle 2}$ times.

The LCM of ${\displaystyle y^1,y^2,y^2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

Multiply ${\displaystyle y}$ by ${\displaystyle y}$.

The LCM for ${\displaystyle 2y,6y^2,6y^2}$ is the numeric part ${\displaystyle 6}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{1}{2y}-\frac{1}{6y^2}=-\frac{7}{6y^2}}$ by ${\displaystyle 6y^2}$.

Simplify the left side.

Simplify the right side.

Move all terms not containing ${\displaystyle y}$ to the right side of the equation.

Divide each term in ${\displaystyle 3y=-6}$ by ${\displaystyle 3}$ and simplify.