Solve for x x+(3x+5)/(x+5)=(2x)/(x+5)
The given problem is an algebraic equation with a variable x. The question requires you to manipulate the terms and solve for the value of x that makes the equation true. It involves algebraic expressions on both sides of the equality sign, with one side having a sum of x and a rational expression, while the other side has a single rational expression. The task is to find the value of x by performing operations such as addition, subtraction, multiplication, division, and the possible simplification of fractions until x is isolated on one side of the equation.
Identify the Least Common Denominator (LCD) for the equation's terms.
To find the LCD, determine the Least Common Multiple (LCM) of the denominators
The LCM is the smallest number that each of the given numbers can divide into without a remainder. To find it: 1. Prime factorize each number. 2. Take the highest power of each prime factor found in any of the numbers.
Since the number
The LCM of
The expression
The LCM of
Clear the fractions by multiplying every term in the equation
Multiply each term by
Simplify the left-hand side of the equation.
Apply simplification to each term.
Use the distributive property:
Multiply
Reorder the multiplication:
Eliminate the common factors of
Remove the common factor:
Rewrite without the fraction:
Combine like terms:
Simplify the right-hand side.
Remove the common factor of
Eliminate the common factor:
Rewrite without the fraction:
Solve the resulting quadratic equation.
Move all terms with
Subtract
Factor the quadratic expression
Find two numbers that multiply to
Write the factors:
Set each factor equal to zero:
Solve for
Set
Subtract
Solve for
Set
Subtract
Combine the solutions to get the final answer:
Check which solutions satisfy the original equation:
Least Common Denominator (LCD): The LCD is the smallest multiple that is common to the denominators of a set of fractions. It is used to combine fractions into a single fraction or to eliminate fractions from an equation.
Least Common Multiple (LCM): The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set. It is found by prime factorization and taking the highest power of each prime factor that appears in any of the numbers.
Prime Factorization: Breaking down a number into its prime factors, which are prime numbers that multiply together to give the original number.
Distributive Property: A property of multiplication over addition or subtraction, stating that
Factoring Quadratics: The process of breaking down a quadratic expression into a product of two binomials. The AC method involves finding two numbers that multiply to give the product of the coefficient of
Zero Product Property: If the product of two factors is zero, then at least one of the factors must be zero. This property is used to solve quadratic equations by setting each factor equal to zero and solving for the variable.
Checking Solutions: After solving an equation, it is important to substitute the solutions back into the original equation to verify that they satisfy the equation. This step ensures that no extraneous solutions are included in the final answer.