Problem 14 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{15}{y-4}+\frac{20}{4-y}\)

Step-by-Step Solution

Answer

\(-\frac{5}{y-4}\)

Step 1: Recognize a Property of Fractions

Notice that the denominators are \(y-4\) and \(4-y\). Recognize that \(4-y = -(y-4)\), meaning these two fractions can be simplified by changing signs of one of the denominators.

Step 2: Rewrite the Second Fraction

Rewrite the second fraction \(\frac{20}{4-y}\) as \(-\frac{20}{y-4}\) by recognizing that \(\frac{1}{4-y} = -\frac{1}{y-4}\).

Step 3: Combine the Fractions

Once both fractions have the same denominator, combine them: \(\frac{15}{y-4} - \frac{20}{y-4}\).

Step 4: Subtract the Numerators

Perform the subtraction in the numerators: \(15 - 20 = -5\). So, you get \(\frac{-5}{y-4}\).

Step 5: Write the Simplified Result

Since we have combined the fractions, the simplified result is \(-\frac{5}{y-4}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is like cleaning up the content of a fraction to make it as simple as possible. We do this to improve readability and make mathematical operations easier. A fraction consists of a numerator (the top number) and a denominator (the bottom number). To simplify, we may need to:

• Remove common factors between the numerator and denominator.
• Recognize equivalent forms.

In the given exercise, the fractions are \( \frac{15}{y-4} \) and \( \frac{20}{4-y} \). Notice that \(4-y = -(y-4)\), reshaping the second fraction allows us to simplify it to a uniform denominator.
This process shows how recognizing relationships between terms helps streamline problem-solving.

Common Denominators

To work easily with fractions, having a common denominator simplifies operations like addition and subtraction. Denominators tell us into how many parts the whole is divided. When fractions have the same denominator, it lets you treat the numerators freely: you can simply add or subtract them.
In the exercise, the fractions \( \frac{15}{y-4} \) and \( \frac{20}{4-y} \) need a common denominator. By identifying that \(4-y = -(y-4)\), we can express both fractions over \(y-4\). This commonality eases the process of combining them into a single fraction, turning potential complexities into a simple arithmetic task.

Fraction Subtraction

Subtracting fractions may initially seem tricky, but once you have a common denominator, the process follows the rules of basic arithmetic. Simply subtract one numerator from the other, keeping the common denominator unchanged.
For the example \( \frac{15}{y-4} - \frac{20}{y-4} \), we subtract the numerators: \(15 - 20 = -5\). The expression simplifies into a single fraction: \( \frac{-5}{y-4} \).
Having accomplished the hard part of getting the denominators to align, all that's left is a straightforward arithmetic operation in the numerators. This makes subtraction much more manageable.

Properties of Fractions

Understanding the properties of fractions is crucial in simplifying and operating on them effectively. One important property of fractions is knowing how flipping signs affects the value. For instance, if you notice that \(b-a\) is simply the negative of \(a-b\), like in the exercise.

• This insight can change how you handle fractions with expressions as denominators.
• It makes manipulation more intuitive and less error-prone.

Recognizing these properties helps enhance mathematical reasoning and permits seamless transitions when dealing with complex fractions. Simplifying expressions becomes a chain reaction of small tweaks and insights, like changing signs to ease computations.