Solve for x 1/(2x)+1/x=1/15

Solution:

Step:1

Identify the least common denominator (LCD) for the fractions in the equation.

Step:1.1

To find the LCD, determine the least common multiple (LCM) of the denominators $2x$, $x$, and $15$.

Step:1.2

The LCM process involves two parts: finding the LCM of the numerical coefficients $2$, $1$, and $15$, and then finding the LCM for the variable parts $x^1$ and $x^1$.

Step:1.3

The LCM is the smallest number that each of the original numbers can divide into without a remainder. The process includes:

1. Listing out the prime factors for each number.

2. Multiplying each prime factor by the highest power it appears with in any of the numbers.

Step:1.4

The number $2$ is already prime.

Step:1.5

The number $1$ is not considered prime as it only has one distinct positive divisor, itself.

Step:1.6

The number $15$ has prime factors $3$ and $5$, or $3\cdot 5$.

Step:1.7

The LCM of $2$, $1$, and $15$ is found by multiplying the prime factors, each to their highest power, resulting in $2\cdot 3\cdot 5$.

Step:1.8

Perform the multiplication of $2\cdot 3\cdot 5$.

Step:1.8.1

First, multiply $2$ by $3$ to get $6\cdot 5$.

Step:1.8.2

Then, multiply $6$ by $5$ to get $30$.

Step:1.9

The variable $x^1$ simplifies to $x$, which appears once.

Step:1.10

The LCM of $x^1$ and $x^1$ is simply $x$, as it is already the highest power of $x$ present.

Step:1.11

Combine the numeric LCM $30$ with the variable part $x$ to get the overall LCD of $30x$.

Step:2

Clear the fractions by multiplying each term in the equation $\frac{1}{2x}+\frac{1}{x}=\frac{1}{15}$ by the LCD $30x$.

Step:2.1

Apply the multiplication to each term: $\frac{30x}{2x}+\frac{30x}{x}=\frac{30x}{15}$.

Step:2.2

Begin simplifying the left side of the equation.

Step:2.2.1

Simplify each fraction individually.

Step:2.2.1.1

Use the commutative property to rearrange the multiplication.

Step:2.2.1.2

Reduce common factors where possible.

Step:2.2.1.2.1

Extract the factor of $2$ from $30$.

Step:2.2.1.2.2

Extract the factor of $2$ from $2x$.

Step:2.2.1.2.3

Cancel out the common factors.

Step:2.2.1.2.4

Rewrite the simplified expression.

Step:2.2.1.3

Combine the coefficient $15$ with $\frac{1}{x}$.

Step:2.2.1.4

Cancel out the common variable $x$.

Step:2.2.1.4.1

Perform the cancellation.

Step:2.2.1.4.2

Rewrite the simplified expression.

Step:2.2.1.5

Apply the commutative property again if necessary.

Step:2.2.1.6

Combine the coefficient $30$ with $\frac{1}{x}$.

Step:2.2.1.7

Cancel out the common variable $x$.

Step:2.2.1.7.1

Perform the cancellation.

Step:2.2.1.7.2

Rewrite the simplified expression.

Step:2.2.2

Add the numbers $15$ and $30$ together.

Step:2.3

Simplify the right side of the equation.

Step:2.3.1

Cancel out the common factor of $15$.

Step:2.3.1.1

Factor out $15$ from $30x$.

Step:2.3.1.2

Perform the cancellation.

Step:2.3.1.3

Rewrite the simplified expression.

Step:3

Solve the simplified equation for $x$.

Step:3.1

Rewrite the equation to isolate the term with $x$.

Step:3.2

Divide both sides of the equation by $2$ to solve for $x$.

Step:3.2.1

Apply the division to both sides of the equation.

Step:3.2.2

Simplify the left side of the equation.

Step:3.2.2.1

Cancel out the common factor of $2$.

Step:3.2.2.1.1

Perform the cancellation.

Step:3.2.2.1.2

Divide $x$ by $1$ to find the value of $x$.

Step:4

Present the solution in various forms.

Exact Form: $x=\frac{45}{2}$

Decimal Form: $x=22.5$

Mixed Number Form: $x=22\frac{1}{2}$

Knowledge Notes:

The problem involves solving a linear equation with fractions. The key steps in solving such an equation include:

1. Finding the Least Common Denominator (LCD): This is crucial for combining fractions by ensuring that all terms have the same denominator. The LCD is the least common multiple (LCM) of the denominators.

2. Clearing the Fractions: By multiplying each term by the LCD, we eliminate the fractions, simplifying the equation.

3. Simplifying the Equation: This involves reducing terms, canceling common factors, and combining like terms.

4. Solving for the Variable: Once the equation is simplified, we isolate the variable on one side to find its value.

5. Prime Factorization: This is used to find the LCM of the numerical parts of the denominators. Prime factorization breaks down a number into its prime factors.

6. Commutative Property of Multiplication: This property states that the order of factors can be changed without affecting the product, which can be useful for rearranging terms during simplification.

7. Cancellation: When a number or variable appears in both the numerator and denominator, it can be canceled out, simplifying the expression.

Understanding these concepts is essential for solving equations with fractions and can be applied to a wide range of algebraic problems.