Problem 26 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{-y+1}{y}-\frac{2 y-5}{3 y}\)

Step-by-Step Solution

Answer

\( \frac{-5y + 8}{3y} \)

Step 1: Identify the Common Denominator

The common denominator for the fractions \( \frac{-y+1}{y} \) and \( \frac{2y-5}{3y} \) is \( 3y \) since it is the least common multiple of \( y \) and \( 3y \).

Step 2: Rewrite Each Fraction with the Common Denominator

Multiply the numerator and denominator of the first fraction by 3 to get \( \frac{3(-y+1)}{3y} \). The second fraction \( \frac{2y-5}{3y} \) already has the correct denominator. Now we have \( \frac{3(-y+1)}{3y} - \frac{2y-5}{3y} \).

Step 3: Combine the Fractions

Since the denominators are the same, combine the numerators: \( \frac{3(-y+1) - (2y-5)}{3y} \).

Step 4: Simplify the Combined Numerator

Distribute the 3 in \( 3(-y+1) \) to get \( -3y + 3 \). Subtract \( (2y - 5) \), which is \( 2y - 5 \). The expression becomes \(-3y + 3 - 2y + 5\).

Step 5: Combine Like Terms in the Numerator

Combine like terms in the numerator: \(-3y - 2y + 3 + 5\) simplifies to \(-5y + 8\). The expression is now \( \frac{-5y + 8}{3y} \).

Step 6: Final Expression

The final expression \( \frac{-5y + 8}{3y} \) is the simplified result of subtracting the given fractions.

Key Concepts

Least Common Denominator

When dealing with fractions, especially in operations like addition and subtraction, finding a common ground is essential for simplification. This common ground is called the **Least Common Denominator (LCD)**. It acts as a unifying base to make the comparison and simplification of fractions possible. In simple terms, for two fractions, the least common denominator is the smallest multiple that can accommodate both denominators without leaving a remainder.
The importance of the LCD is evident when you want everything to "speak the same language." For instance, with the fractions \( \frac{-y+1}{y} \) and \( \frac{2y-5}{3y} \), the denominators are \( y \) and \( 3y \), respectively. The smallest multiple they share is \( 3y \), which is crucial for combining these fractions without any mathematical errors.
To find the LCD:

• List the multiples of each denominator.
• Identify the smallest multiple common to both denominators.
• Convert each fraction to have this common denominator.

Finding the LCD is like setting the stage for all fractions to perform together seamlessly!

Fraction Operations

Once you have the least common denominator, the next step is performing operations on the fractions. Fraction operations include addition, subtraction, multiplication, and division, but we'll focus on subtraction here. To subtract fractions efficiently, you must have the same denominator, which is already achieved through the LCD.
Consider the fractions \( \frac{3(-y+1)}{3y} \) and \( \frac{2y-5}{3y} \). With both fractions sharing the common denominator \( 3y \), subtraction is straightforward since only the numerators differ.
Let's walk through the subtraction:

• Keep the common denominator the same.
• Subtract the numerators: \( 3(-y+1) - (2y-5) \).
• Focus solely on the top numbers, viewing it as a single arithmetic problem.

Subtracting fractions is much like playing with building blocks; once the bases (denominators) are the same, moving around the blocks on top (numerators) is easy!

Simplifying Expressions

After performing fraction operations, there's often the need to simplify the expression that results. Simplifying expressions brings clarity and order, reducing complexity to its simplest form. For our problem \( \frac{-5y + 8}{3y} \), we've reached an expression we can work with, but let's refine it further.
Simplification involves:

• Combining like terms: Look for terms that have the same variable parts and combine their coefficients.
• Perform arithmetic operations: Execute any arithmetic, such as subtraction and addition within the numerator.
• Check for common factors in the numerator and the denominator to see if they can be divided out.

For the expression \( -3y + 3 - 2y + 5 \), combining the like terms results in \(-5y + 8\), a much cleaner form.
This step not only simplifies the expression, making it more recognizable and easier to interpret, but also optimizes the solution, ensuring it remains mathematically correct and efficient for anyone to understand.