Problem 46 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{1}{2 x+2 y}-\frac{y}{x+y}\)

Step-by-Step Solution

Answer

\( \frac{1 - 2y}{2(x+y)} \)

Step 1: Find a Common Denominator

To combine the fractions \( \frac{1}{2x + 2y} \) and \( \frac{y}{x + y} \), we need to find a common denominator. The denominators are \( 2x + 2y \) and \( x + y \). Observe that \( 2x + 2y = 2(x+y) \). Thus, the least common denominator (LCD) is \( 2(x+y) \).

Step 2: Adjust Each Fraction to Have the Common Denominator

Rewrite each fraction with the common denominator \( 2(x+y) \):- For \( \frac{1}{2x + 2y} \), multiply the numerator and denominator by 1 to get \( \frac{1}{2(x+y)} \).- For \( \frac{y}{x+y} \), multiply the numerator and the denominator by 2 to get \( \frac{2y}{2(x+y)} \).

Step 3: Subtract the Fractions

Now that both fractions have the same denominator, subtract the fractions:\[ \frac{1}{2(x+y)} - \frac{2y}{2(x+y)} = \frac{1 - 2y}{2(x+y)} \]

Step 4: Simplify the Fraction if Possible

The fraction \( \frac{1 - 2y}{2(x+y)} \) cannot be simplified further. Ensure no further common factors exist in the numerator and denominator.

Key Concepts

Fractions

Fractions are a way to express a part of a whole. They consist of a numerator, placed above the line, and a denominator, placed below. The numerator tells us how many parts we are interested in, while the denominator indicates the total number of equal parts the whole is divided into. Fractions can describe real-world situations, like slicing a cake: if a cake is divided into 8 pieces and you take 3, your portion can be represented as \( \frac{3}{8} \). There are different types of fractions, including:

• Proper fractions: The numerator is smaller than the denominator, like \( \frac{3}{8} \).
• Improper fractions: The numerator is greater or equal to the denominator, such as \( \frac{9}{8} \).
• Mixed numbers: A whole number plus a fraction, for example, 1\( \frac{1}{8} \).

Understanding fractions is essential in algebra, as they often appear in equations and expressions. Simplifying fractions or finding equivalent fractions is a common task to make the math easier to handle.

Common Denominator

When dealing with fractions in algebra, a common denominator is crucial for adding or subtracting fractions. The common denominator is simply a shared multiple between the different denominators of the fractions involved. For example, if you want to add or subtract \( \frac{1}{4} \) and \( \frac{1}{6} \), you notice that 12 is a common multiple of both 4 and 6. Thus, the least common denominator (LCD) for these fractions would be 12. Finding the LCD:

• Identify the denominators you are working with.
• Find the least common multiple (LCM) of these denominators.
• Use this LCM as your common denominator to add or subtract the fractions.

In our exercise, the denominators were \(2x + 2y\) and \(x + y\). By factoring, we identified \(2(x + y)\) as the common denominator. This allowed us to standardize the fractions under a single denominator, making subtraction straightforward.

Subtracting Fractions

Subtracting fractions involves a few steps to ensure they are manipulative in their terms, especially when dealing with variables. Firstly, ensure both fractions share a common denominator. As we saw, this common denominator simplifies the subtraction process. The steps involved in subtracting fractions are:

• Find a common denominator.
• Adjust each fraction to have this common denominator.
• Subtract the numerators, keeping the new denominator the same.

For example, with the fractions \( \frac{1}{2(x+y)} \) and \( \frac{2y}{2(x+y)} \), they share the common denominator \(2(x+y)\). After this setup, you can subtract the numerators: \[ 1 - 2y \]. So, the resulting fraction is \( \frac{1 - 2y}{2(x+y)} \). Remember, always check if your final fraction can be simplified further by finding common factors in the numerator and denominator. If not, the fraction is in its simplest form, as with the exercise result.