Problem 32 - Elementary and Intermediate Algebra

Question

Subtract and simplify the result, if possible. \(\frac{y+8}{2 y(y-14)}-\frac{y}{2 y(y-14)}\)

Step-by-Step Solution

Answer

The simplified result is \(\frac{8}{2y(y-14)}\).

Step 1: Identify the Common Denominator

The denominators of both fractions are already identical: \(2y(y-14)\). This means we do not need to find a new common denominator and can directly subtract the numerators.

Step 2: Subtract the Numerators

Write the expression as a single fraction by subtracting the numerators: \(\frac{y+8 - y}{2y(y-14)}\). The \(y\) terms in the numerator cancel out.

Step 3: Simplify the Expression

Simplify the numerator: \(y+8-y = 8\). Rewrite the expression as \(\frac{8}{2y(y-14)}\).

Step 4: Simplify Further if Possible

Check if the fraction can be simplified further. Since 8 and the terms in the denominator have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Subtracting Fractions

Subtracting fractions may seem a bit tricky at first, but it's no different than subtracting whole numbers when you have a good grasp on the steps. When the denominators of the fractions you are working with are the same, subtracting them becomes much easier. Lucky for us, in the exercise given, both fractions share the denominator \(2y(y-14)\).

When subtracting fractions with identical denominators, you simply need to:

• Keep the common denominator,
• Subtract the numerators.

In our example, subtract \(\frac{y}{2y(y-14)}\) from \(\frac{y+8}{2y(y-14)}\). The expression becomes \(\frac{y+8-y}{2y(y-14)}\). The \(y\) terms in the numerator cancel out, making your life so much easier! This means the subtraction part is done, simplifying our expression significantly!

Simplifying Fractions

Simplifying fractions is all about reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This makes the fraction easier to work with and understand. When you subtract the fractions in our example, you are left with \(\frac{8}{2y(y-14)}\).

First, observe the numerator and the denominator. Notice that 8 is already a whole number and doesn't share any common factors with the terms in the denominator. This tells us that 8 cannot be reduced further with the denominator. As there are no common factors, the fraction is already simplified.
A quick check always helps:

• Are there any common factors other than 1?
• Is the numerator 1, or is it as low as possible?

These questions help ensure that your fraction is in its simplest form.

Common Denominators

Understanding common denominators is a key part of working with fractions, especially when you're adding or subtracting them. A common denominator is simply a shared multiple of the denominators of two or more fractions. In our problem, the fractions both have the denominator \(2y(y-14)\), which makes them "like" fractions.

• This same denominator means you do not need to do additional work to find a common ground—it’s already there.
• With common denominators, subtraction, or addition like in our case becomes much easier.

Always check your denominators:

• If they are the same, just move on to the numerators.
• If they differ, find a common denominator by identifying the Least Common Multiple (LCM).

Having the same or finding a common denominator may seem tedious but it guarantees a seamless subtraction or addition process!