Question

Literal Fractional Equations. $$\frac{x}{a-1}-\frac{x}{a+1}=b$$

Short answer

\(x=\frac{b(a^2-1)}{2}\)

Step-by-step solution

Find a common denominator

To combine the fractions, you need to find a common denominator. The denominators here are \(a-1\) and \(a+1\), so the common denominator will be \(a-1)(a+1)\), which is the product of the two.

Rewrite the fractions with the common denominator

Express each fraction with the common denominator \(a-1)(a+1)\). This gives us \(\frac{x(a+1)}{(a-1)(a+1)}-\frac{x(a-1)}{(a-1)(a+1)}=b\).

Combine the fractions

Now that the fractions have a common denominator, combine them by subtracting the numerators: \(\frac{x(a+1)-x(a-1)}{(a-1)(a+1)}=b\).

Simplify the numerator

Expand and combine like terms in the numerator to simplify. \(x(a+1)-x(a-1) = xa+x- xa+x = 2x\), which leads to \(\frac{2x}{(a-1)(a+1)}=b\).

Clear the fraction

Multiply both sides by the common denominator to get rid of the fraction: \(2x=b(a-1)(a+1)\).

Expand the right side

Expand the right side to simplify the equation further: \(2x = b(a^2-1)\), because \(\left(a-1\right)\left(a+1\right)=a^2-1^2=a^2-1\).

Solve for x

Divide both sides by 2 to solve for \(x\): \(x=\frac{b(a^2-1)}{2}\).

Key concepts

Common Denominator

Understanding the concept of a common denominator is crucial when working with literal fractional equations. A common denominator is a shared multiple of the denominators of two or more fractions, allowing you to combine or compare them. It's like finding a common language for two people who speak different languages.

For instance, consider the denominators in our exercise, \(a-1\) and \(a+1\). The least common denominator (LCD) is their product, \((a-1)(a+1)\). Think of it as the least common ground on which both original fractions can comfortably 'stand' without losing their value.

• It makes it easier to perform addition or subtraction with the fractions.
• This step reduces the complexity of the problem.
• The common denominator is usually the product of the distinct denominators, however, in cases where the denominators have common factors, the LCD is the smallest multiple that includes all factors of both denominators.

Visualize a pizza party where everyone needs equal slices; having a common denominator is like ensuring every slice is the same size so everyone gets their fair share.

Simplifying Fractions

Simplifying fractions is a fundamental skill in algebra that often involves reducing the numerator and denominator to their smallest values. However, in the context of algebraic equations with literals, 'simplifying' takes on additional meaning. It also involves the process of expanding and combining like terms to turn a complicated expression into a simpler one, just like tidying a cluttered room into a state of order.

When we simplified the numerator in our step-by-step solution, we expanded the terms and combined like ones:\(xa+x-xa+x=2x\). Here’s the simplification breakdown:

• Expanding each term involves distributing the variable across the terms inside the brackets.
• Combining like terms then condenses the expression into the simplest form possible.
• It's important to consistently apply the distributive property and watch for positive and negative signs when simplifying.
• This process reduces the fraction to its simplest form, where the numerator and denominator have no common factors other than 1.

The goal here is to make the equation manageable and prepare it for the final steps of solving the equation.

Solving Algebraic Equations

Solving algebraic equations is the endgame of any algebraic exercise; it’s where we find the value(s) that satisfy the equation. With literal fractional equations, the objective is to isolate the variable and solve for it.

From our example, after simplifying the fractional equation to get \(2x = b(a^2-1)\), we got rid of the fraction by multiplying both sides by the common denominator. This is a vital strategy, as it transforms the equation into a simpler format without fractions – effectively 'clearing' the fraction. Subsequent steps generally involve:

• Performing arithmetic operations such as multiplication or division to both sides to isolate the variable.
• Paying attention to the rules of algebra when moving terms from one side to the other.
• Being cautious to perform the same operation on every term to maintain the balance of the equation.

At the end - the final flourish - we divide by 2 to solve for \(x\), obtaining \(x=\frac{b(a^2-1)}{2}\). Solving algebraic equations is a bit like a dance, with steps that need to be followed in sequence to reach the elegant conclusion of finding the value of the unknown variable.