Problem 95 - Elementary Algebra

Question

For the following problems, convert the given rational expressions to rational expressions having the same denominators. $$ \frac{k}{5-k}, \frac{3 k}{k-5} $$

Step-by-Step Solution

Answer

#tag_title#Short Answer#tag_content# The rational expressions with a common denominator of \((5-k)\) are: \(\frac{k}{5-k}\) and \(\frac{-3k}{5-k}\).

Step 1: Identify the current denominators

In the given rational expressions, the denominators are \((5-k)\) and \((k-5)\).

Step 2: Find the common denominator

Notice that \((5-k)\) and \((k-5)\) are related, as \((k-5) = -1(5-k)\). Thus, the common denominator for both fractions can be \((5-k)\) or \((k-5)\). For simplicity, we will choose \((5-k)\) as the common denominator.

Step 3: Convert the given rational expressions to have a common denominator

First, let's rewrite the first rational expression, since its denominator is already \((5-k)\): $$ \frac{k}{5-k} $$ Now let's rewrite the second rational expression: $$ \frac{3k}{k-5} = \frac{3k}{ -1(5-k)} = \frac{-3k}{5-k} $$ The converted rational expression is: $$ \frac{-3k}{5-k} $$ So the given rational expressions, now with the same denominators, are: $$ \frac{k}{5-k} , \frac{-3k}{5-k} $$

Key Concepts

Common Denominator

When working with rational expressions, just like with numerical fractions, we sometimes need to combine them. To do this effectively, these rational expressions must have the same base or denominator. This shared base is known as the common denominator.
Finding a common denominator is especially important in algebra, where fractions may involve variables. In the given problem, you may notice that the denominators are \(5-k\) and \(k-5\). They look quite similar but aren't identical. However, they are multiplicatively related, as \(k-5 = -1(5-k)\). This allows us to choose either as the common denominator, but for ease, \(5-k\) is often selected.

• Identify denominators in each rational expression.
• Determine a common denominator, simplifying when possible.

Finding a common denominator makes addition, subtraction, and comparison of algebraic fractions straightforward. It transforms different-looking fractions into ones that are much simpler to handle in further algebraic operations.

Algebraic Fractions

Algebraic fractions are fractions that include variables in their numerators, denominators, or both. They function under the same rules as regular fractions but require handling of the algebraic expressions involved. In our problem, \(\frac{k}{5-k}\) and \(\frac{3k}{k-5}\) are typical examples of algebraic fractions.
It's crucial to recognize that these fractions can be manipulated using algebra rules. For instance, you might multiply or divide both numerator and denominator by the same expression to find common denominators or simplify.

• Ensure that you're aware of restrictions, like values that could make a denominator zero.
• Use identities and algebraic properties to adjust the fractions as required.

Handling algebraic fractions efficiently can significantly ease solving equations, factoring expressions, and performing complex algebraic operations. As you become more familiar with them, these fractions become less intimidating and more of a regular tool in your algebra toolkit.

Fractions with Variables

Fractions with variables, also known as variable fractions, are an extension of basic fractions that involve variables and constants. Similar to numerical fractions, they express a part of a whole, but variables add complexity as their values can change.
In our example, both \(\frac{k}{5-k}\) and \(\frac{3k}{k-5}\) involve variables, making it crucial to handle them with care. Identifying expressions like \(5-k\) and \(k-5\) helps you see how they relate to each other, enabling simplification or conversion.

• Treat any variable in the denominator with caution, as its value can't make the denominator zero to keep the expression defined.
• Understand properties like the need for common denominators when adding or subtracting variable fractions.

Working with fractions that include variables demands a solid grasp of algebra basics. Ensure you apply algebraic principles consistently, and remember that practice makes perfect in mastering these fractions.