Problem 13 - Elementary Algebra

Question

Simplify each complex rational expression. $$ \frac{1-\frac{1}{y}}{1+\frac{1}{y}} $$

Step-by-Step Solution

Answer

Question: Simplify the complex rational expression given by: $$\frac{1-\frac{1}{y}}{1+\frac{1}{y}}$$. Answer: The simplified form of the given complex rational expression is: $$\frac{y-1}{y+1}.$$

Step 1: Multiply by the least common denominator (LCD)

In order to eliminate fractions, we'll multiply both the numerator and the denominator by the least common denominator of both of them, which is y. $$ \frac{1-\frac{1}{y}}{1+\frac{1}{y}} \times \frac{y}{y} $$

Step 2: Distribute y

Next, we'll distribute y to both the numerator and the denominator: $$ \frac{y(1-\frac{1}{y})}{y(1+\frac{1}{y})} $$

Step 3: Simplify

Now, we'll simplify the numerator and the denominator: $$ \frac{y(1)-y(\frac{1}{y})}{y(1)+y(\frac{1}{y})} $$ $$ \frac{y-\cancel{y}\cdot\frac{1}{\cancel{y}}}{y+\cancel{y}\cdot\frac{1}{\cancel{y}}} $$ $$ \frac{y-1}{y+1} $$ So, the simplified form of the given complex rational expression is: $$ \frac{y-1}{y+1} $$

Key Concepts

Simplifying Rational Expressions

When it comes to simplifying rational expressions, one of the key goals is to make complex expressions more manageable and easier to understand. Simplification can include a range of techniques, from factoring to finding common denominators. The idea is to transform a complicated expression into a simpler one without changing its value.

To achieve this, you may need to do a variety of things such as:

• Identify and cancel common factors in the numerator and denominator.
• Multiply by the least common denominator to eliminate complex fractions within fractions.
• Apply algebraic operations to combine or separate terms.

Least Common Denominator

The least common denominator (LCD) is crucial when working with complex fractions. It is the smallest number that both denominators can divide into without a remainder. In other words, it's the least common multiple of the denominators involved.

How to find the LCD:

• List the multiples of each denominator.
• Identify the smallest multiple common to both lists.
• Use this common multiple as your LCD to combine fractions.

The process smoothens the path to combine fractions and simplifies the overall expression.

Algebraic Fraction Simplification

Algebraic fraction simplification is about reducing algebraic expressions within fractions to their simplest form. This means handling not only numbers but also variables. The process often involves multiple steps:

• Reduce fractions inside of the main fraction (if present).
• Find the least common denominator for the entire expression.
• Combine terms methodically and cancel where possible.

Remember that it's vital to perform the same operations on both the numerator and the denominator to keep the expression equivalent to the original.

Distributive Property

The distributive property is a key player in simplifying algebraic expressions. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then summing the products. For instance, in the expression \(a(b+c)\), you would distribute the \(a\) to both \(b\) and \(c\), resulting in \(ab + ac\).

In our original problem, this property comes into play when we multipliy the common denominator through the numerator and the denominator, which helps to get rid of the more complex fraction within a fraction, streamlining the simplification process.