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

Solution:

Step:1

Isolate terms with $x$ on one side of the equation.

Step:1.1

Subtract $\frac{4}{7}$ from both sides: $\frac{3}{x}=\frac{2x+1}{7x}-\frac{4}{7}$

Step:1.2

Simplify the equation.

Step:1.2.1

Divide $\frac{2x+1}{7x}$ into two parts: $\frac{3}{x}=\frac{2x}{7x}+\frac{1}{7x}-\frac{4}{7}$

Step:1.2.2

Reduce fractions by canceling $x$.

Step:1.2.2.1

Eliminate $x$ where possible: $\frac{3}{x}=\frac{2}{7}+\frac{1}{7x}-\frac{4}{7}$

Step:1.2.2.2

Rewrite the equation: $\frac{3}{x}=\frac{2}{7}+\frac{1}{7x}-\frac{4}{7}$

Step:1.3

Combine like terms over a common denominator: $\frac{3}{x}=\frac{2-4}{7}+\frac{1}{7x}$

Step:1.4

Subtract $4$ from $2$: $\frac{3}{x}=\frac{-2}{7}+\frac{1}{7x}$

Step:1.5

Place the negative sign in front: $\frac{3}{x}=-\frac{2}{7}+\frac{1}{7x}$

Step:2

Determine the least common denominator (LCD) of the equation.

Step:2.1

The LCD is the least common multiple (LCM) of the denominators: $x,7,7x$

Step:2.2

To find the LCM, separate the process for numbers and variables: LCM of $1,7,7$ and LCM of $x^1,x^1$.

Step:2.3

LCM is the smallest number divisible by all numbers: List prime factors and multiply the highest power of each.

Step:2.4

The number $1$ is not prime.

Step:2.5

Since $7$ is prime, it remains unchanged.

Step:2.6

The LCM of $1,7,7$ is $7$.

Step:2.7

The variable $x^1$ simplifies to $x$.

Step:2.8

The LCM of $x^1,x^1$ is $x$.

Step:2.9

Combine the numeric and variable parts for the LCD: $7x$

Step:3

Eliminate fractions by multiplying each term by the LCD, $7x$.

Step:3.1

Multiply the equation by $7x$: $\frac{3}{x}(7x)=-\frac{2}{7}(7x)+\frac{1}{7x}(7x)$

Step:3.2

Simplify the left side.

Step:3.2.1

Use the commutative property: $7\frac{3}{x}x=-\frac{2}{7}(7x)+\frac{1}{7x}(7x)$

Step:3.2.2

Multiply $7$ and $\frac{3}{x}$: $\frac{21}{x}x=-\frac{2}{7}(7x)+\frac{1}{7x}(7x)$

Step:3.2.3

Cancel $x$: $21=-\frac{2}{7}(7x)+\frac{1}{7x}(7x)$

Step:3.3

Simplify the right side.

Step:3.3.1

Simplify each term.

Step:3.3.1.1

Cancel $7$: $21=-2x+\frac{1}{7x}(7x)$

Step:3.3.1.2

Use the commutative property: $21=-2x+7\frac{1}{7x}x$

Step:3.3.1.3

Cancel $7$: $21=-2x+\frac{1}{x}x$

Step:3.3.1.4

Cancel $x$: $21=-2x+1$

Step:4

Solve for $x$.

Step:4.1

Rearrange the equation: $-2x+1=21$

Step:4.2

Isolate $x$-terms.

Step:4.2.1

Subtract $1$ from both sides: $-2x=21-1$

Step:4.2.2

Calculate the result: $-2x=20$

Step:4.3

Divide by $-2$ to solve for $x$.

Step:4.3.1

Divide each term by $-2$: $\frac{-2x}{-2}=\frac{20}{-2}$

Step:4.3.2

Simplify the left side: $x=\frac{20}{-2}$

Step:4.3.3

Simplify the right side: $x=-10$

Knowledge Notes:

1. Isolating Variables: When solving for a variable, it's common to isolate terms containing the variable on one side of the equation.

2. Simplifying Fractions: Fractions can be simplified by canceling out common factors in the numerator and denominator.

3. Least Common Denominator (LCD): The LCD of several fractions is the least common multiple (LCM) of their denominators. It's used to combine fractions or eliminate them from an equation.

4. Prime Numbers: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Prime factorization helps in finding the LCM.

5. Commutative Property of Multiplication: This property states that the order in which two numbers are multiplied does not affect the product.

6. Solving Linear Equations: Linear equations are solved by performing operations that maintain the equality, such as adding, subtracting, multiplying, or dividing both sides by the same number.