Solve for x 4/7+3/x=(2x+1)/(7x)

Subtract ${\displaystyle \frac{4}{7}}$ from both sides of the equation.

Simplify each term.

Combine the numerators over the common denominator.

Subtract ${\displaystyle 4}$ from ${\displaystyle 2}$.

Move the negative in front of the fraction.

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

Since ${\displaystyle x,7,7x}$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part ${\displaystyle 1,7,7}$ then find LCM for the variable part ${\displaystyle x^1,x^1}$.

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.

Since ${\displaystyle 7}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 7}$.

The LCM of ${\displaystyle 1,7,7}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

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

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

The LCM for ${\displaystyle x,7,7x}$ is the numeric part ${\displaystyle 7}$ multiplied by the variable part.

Multiply each term in ${\displaystyle \frac{3}{x}=-\frac{2}{7}+\frac{1}{7x}}$ by ${\displaystyle 7x}$.

Simplify the left side.

Simplify the right side.

Rewrite the equation as ${\displaystyle -2x+1=21}$.

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

Divide each term in ${\displaystyle -2x=20}$ by ${\displaystyle -2}$ and simplify.