Question

Identify the least common denominator of the equation. $$\frac{5 x}{x-1}-\frac{7}{x}=\frac{9}{x}$$

Short answer

The least common denominator is \(x(x-1)\).

Step-by-step solution

Identify the Denominators

The denominators in the equation are \(x-1\) and \(x\).

Determine the Least Common Denominator

The least common denominator (LCD) of a set of fractions is the smallest expression that each of the denominators can divide into. The LCD will therefore be the multiple of the unique factors of each denominator, which in this case will be \(x(x-1)\).

Key concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. It provides a clear, concise way to represent mathematical thoughts and the flexibility to solve a variety of problems.

In algebra, we work with unknown values called variables, represented by letters like 'x' and 'y'. Operations such as addition, subtraction, multiplication, and division can be performed on these variables, just like with numbers.

In the case of finding the least common denominator (LCD) in algebraic equations, algebra simplifies the process of combining, comparing, and solving rational expressions and equations. Understanding basic algebraic principles is essential to master this skill.

Rational Equations

Rational equations are equations containing rational expressions, which are fractions that have polynomials in the numerator, the denominator, or both. Solving these types of equations often requires finding a common denominator.

A common issue when working with rational equations is dealing with undefined expressions; for instance, when a denominator equals zero, the expression is undefined. It's vital to recognize and exclude these values from the solution set since they make the equation invalid.

In the given exercise, the equation involves rational expressions with different denominators. To solve it effectively, you need to understand how to manipulate and combine these expressions, which is a fundamental skill in working with rational equations.

LCD in Algebra

In algebra, 'Least Common Denominator' (LCD) refers to the least common multiple of all denominators present in a set of fractions or rational expressions. It’s the smallest expression into which each denominator can evenly divide. This concept allows us to combine fractions into a single expression or equation.

The process of finding the LCD typically involves factoring each denominator to find the prime numbers or variables that constitute it and then taking the highest power of each unique factor to form the LCD. In the exercise provided, the denominators are \( x-1 \) and \( x \). To find the LCD, we look for the product of the distinct factors, which gives us \( x(x-1) \), as this will be divisible by each individual denominator.

Remember, using the LCD to combine fractions is a crucial step before solving the equation. Once combined, the equation can often be simplified to a more familiar linear or quadratic equation and solved using standard algebraic methods.