Problem 12 - Algebra 2 and Trigonometry

Question

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ 5-\frac{1}{2 y} $$

Step-by-Step Solution

Answer

The simplest form is \( \frac{10y - 1}{2y} \), undefined for \( y = 0 \).

Step 1: Identify the Terms

We are given the expression \( 5 - \frac{1}{2y} \). It involves a subtraction between the whole number 5 and the fraction \( \frac{1}{2y} \). Our goal is to perform the subtraction and simplify the result.

Step 2: Common Denominator Setup

Since we have a whole number and a fraction, we need to express 5 as a fraction with the same denominator as \( \frac{1}{2y} \). So, rewrite 5 as \( \frac{10y}{2y} \). Now both terms have a denominator of \( 2y \).

Step 3: Subtract Fractions

Now subtract the fractions: \( \frac{10y}{2y} - \frac{1}{2y} = \frac{10y - 1}{2y} \). This is the effective subtraction in a single fraction.

Step 4: Simplify the Fraction

The result of the subtraction is \( \frac{10y - 1}{2y} \). This fraction is already in its simplest form because there are no common factors in the numerator and denominator besides 1.

Step 5: Identify Undefined Values

A fraction is undefined if its denominator equals zero. Here, the denominator is \( 2y \), so we solve \( 2y = 0 \). This gives \( y = 0 \) as the value where the fraction is undefined.

Key Concepts

Subtraction of Fractions

Subtraction of fractions can seem tricky because you often deal with both whole numbers and fractions together. In our exercise, we began with the expression \( 5 - \frac{1}{2y} \). To successfully subtract, both numbers need to be expressed as fractions with a common denominator. This ensures that the subtraction is done uniformly across/between the numerators.When you subtract fractions, always remember:

• Convert all terms involved to have a common denominator.
• Only the numerators need to be subtracted; the denominator remains unchanged.
• Simplify the result if possible once you've done the subtraction.

Common Denominator

Having a common denominator is crucial for adding or subtracting fractions. In our scenario with \( 5 - \frac{1}{2y} \), the fraction \( \frac{1}{2y} \) already had a denominator of \( 2y \). The whole number 5 is rewritten as \( \frac{10y}{2y} \), which has the same denominator.This concept is all about aligning the base (denominator) of two or more fractions so they can be subtracted or added together easily:

• Rewrite integers as fractions with the denominator of concern. In our example, 5 is rewritten by finding an equivalent fraction.
• Once equivalent fractions are found, subtraction is straightforward between the numerators while keeping the denominator constant.

Undefined Values

Undefined values in fractions occur whenever the denominator becomes zero because division by zero is mathematically undefined. In the expression \( \frac{10y - 1}{2y} \), the denominator is \( 2y \). To pinpoint when the fraction becomes undefined:

• Set the denominator equal to zero: \( 2y = 0 \).
• Solve for the variable: \( y = 0 \) will result in an undefined fraction.

Recognizing undefined values helps prevent mathematical errors and ensures that all solutions consider the full range of variable legality.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. In our example, the fraction \( \frac{10y - 1}{2y} \) cannot be simplified further.To know when a fraction is in its simplest form:

• Check if the numerator and the denominator have any common factors (other than 1).
• If they do, divide both the numerator and the denominator by these common factors.
• If they don’t, like in our case, then the fraction is already simplified.

Simplifying is important for clarity in mathematics, as it ensures solutions are presented in the neatest possible form.