Solve for x 1/(2x)+1/x=1/15
The question is asking for the correct value of 'x' that satisfies the equation provided. Specifically, the equation is a rational equation where the variable 'x' appears in the denominators of the fractions. You will need to find the common denominator, combine the fractions, and then solve for 'x' through algebraic means, often involving multiplying both sides of the equation to get rid of the fractions and isolating 'x' on one side of the equation.
Identify the least common denominator (LCD) for the fractions in the equation.
To find the LCD, determine the least common multiple (LCM) of the denominators
The LCM process involves two parts: finding the LCM of the numerical coefficients
The LCM is the smallest number that each of the original numbers can divide into without a remainder. The process includes:
Listing out the prime factors for each number.
Multiplying each prime factor by the highest power it appears with in any of the numbers.
The number
The number
The number
The LCM of
Perform the multiplication of
First, multiply
Then, multiply
The variable
The LCM of
Combine the numeric LCM
Clear the fractions by multiplying each term in the equation
Apply the multiplication to each term:
Begin simplifying the left side of the equation.
Simplify each fraction individually.
Use the commutative property to rearrange the multiplication.
Reduce common factors where possible.
Extract the factor of
Extract the factor of
Cancel out the common factors.
Rewrite the simplified expression.
Combine the coefficient
Cancel out the common variable
Perform the cancellation.
Rewrite the simplified expression.
Apply the commutative property again if necessary.
Combine the coefficient
Cancel out the common variable
Perform the cancellation.
Rewrite the simplified expression.
Add the numbers
Simplify the right side of the equation.
Cancel out the common factor of
Factor out
Perform the cancellation.
Rewrite the simplified expression.
Solve the simplified equation for
Rewrite the equation to isolate the term with
Divide both sides of the equation by
Apply the division to both sides of the equation.
Simplify the left side of the equation.
Cancel out the common factor of
Perform the cancellation.
Divide
Present the solution in various forms.
Exact Form:
Decimal Form:
Mixed Number Form:
The problem involves solving a linear equation with fractions. The key steps in solving such an equation include:
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.
Clearing the Fractions: By multiplying each term by the LCD, we eliminate the fractions, simplifying the equation.
Simplifying the Equation: This involves reducing terms, canceling common factors, and combining like terms.
Solving for the Variable: Once the equation is simplified, we isolate the variable on one side to find its value.
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.
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.
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.