Solve for a (a+1)/(5a)-1/a=1

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

Since ${\displaystyle 5a,a,1}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 5,1,1}$ then find LCM for the variable part ${\displaystyle a^1,a^1}$.

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

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Since ${\displaystyle 5}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 5}$.

${\displaystyle 5}$ is a prime number

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

Not prime

The LCM of ${\displaystyle 5,1,1}$ 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.

${\displaystyle a}$ occurs ${\displaystyle 1}$ time.

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

The LCM for ${\displaystyle 5a,a,1}$ is the numeric part ${\displaystyle 5}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{a+1}{5a}-\frac{1}{a}=1}$ by ${\displaystyle 5a}$.

Simplify the left side.

Simplify the right side.

Move all terms containing ${\displaystyle a}$ to the left side of the equation.

Add ${\displaystyle 4}$ to both sides of the equation.

Divide each term in ${\displaystyle -4a=4}$ by ${\displaystyle -4}$ and simplify.