Problem 91 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{4 a}{a-5}+a$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{a(a-1)}{a-5} \).

Step 1: Identify the common denominator

The given expression is \( \frac{4a}{a-5} + a \). To add these two terms, it is important to have a common denominator. The term \( a \) needs to be rewritten with the denominator \( a-5 \).

Step 2: Rewrite the second term

The second term \( a \) can be rewritten as \( \frac{a(a-5)}{a-5} \) so that it has the same denominator as the first term. Thus, \( \frac{4a}{a-5} + \frac{a(a-5)}{a-5} \).

Step 3: Combine the fractions

Add the numerators together since they now have a common denominator: \( \frac{4a + a(a - 5)}{a-5} \).

Step 4: Simplify the expression

Expand the term \( a(a-5) \) to get \( a^2 - 5a \). Substitute back to get \( \frac{4a + a^2 - 5a}{a-5} \).

Step 5: Simplify the numerator

Combine like terms in the numerator: \( \frac{a^2 - a}{a-5} \).

Step 6: Factor the numerator

Factor out \( a \) from the numerator: \( \frac{a(a-1)}{a-5} \).

Key Concepts

Addition of Rational Expressions

Adding rational expressions involves combining two or more algebraic expressions that are in fraction form. The process is similar to adding numerical fractions. First, you must ensure that all expressions have a common denominator. Without a common denominator, you can't directly add the numerators. This is why it's essential to align them properly before proceeding.

In the example given, the expression is \( \frac{4a}{a-5} + a \). The first term is already in fraction form, whereas the second term, \( a \), is not. To add these rational expressions, the second term needs an appropriate form that includes the common denominator \( a-5 \). This is a crucial step because it allows for a straightforward addition of the numerators.

Common Denominator

Finding a common denominator ensures that both terms in a rational expression are on equal footing. Think of it as finding a common language between parts of a fraction. A common denominator helps in smoothly combining or comparing different fractions.

For the expression \( \frac{4a}{a-5} + a \), the common denominator is \( a-5 \). By changing \( a \) into a fraction with this common denominator, you adjust the problem for seamless addition. The second term \( a \) is rewritten as \( \frac{a(a-5)}{a-5} \). This maneuver aligns both terms, allowing the numerators to be added together, setting the stage for further simplifying.

Simplifying Rational Expressions

Once a common denominator is established and the terms are combined, simplifying the rational expression becomes straightforward. Simplification often involves combining like terms and canceling out factors if possible. This results in a tidier expression.

Combining the numerators \( 4a \) and \( a(a-5) \), they simplify to \( 4a + a^2 - 5a \), which further reduces to \( a^2 - a \) when combining like terms. The simplified expression \( \frac{a^2 - a}{a-5} \) is much easier to interpret and work with in successive calculations or evaluations.

Factoring in Algebra

Factoring is a powerful algebraic tool used to simplify expressions. It involves breaking down complex expressions into products of simpler factors. This technique often reveals opportunities to simplify rational expressions further.

In the rational expression \( \frac{a^2 - a}{a-5} \), the numerator \( a^2 - a \) can be factored by taking out the common factor \( a \). This results in \( a(a-1) \), creating a new expression \( \frac{a(a-1)}{a-5} \). While this example does not allow for further reduction since the denominator does not share any factors with \( a(a-1) \), factoring still delivers a simpler and more readable final expression. Being adept at factoring can save effort and time when working with intricate rational expressions.