Problem 14 - Elementary Algebra

Question

Add or subtract as indicated, and express your answers in lowest terms. (Objective 1) $$\frac{17}{y}+\frac{12}{y}$$

Step-by-Step Solution

Answer

\( \frac{29}{y} \)

Step 1: Combine the Fractions

The fractions have the same denominator, which is \( y \). When the denominators are the same, you can combine the fractions by adding the numerators. Therefore, \( \frac{17}{y} + \frac{12}{y} = \frac{17+12}{y} \).

Step 2: Add the Numerators

Now, add the numerators: \( 17 + 12 = 29 \). Substitute this sum back into the fraction to combine the expression: \( \frac{29}{y} \).

Step 3: Simplify the Expression

Check if the numerator and the denominator have any common factors. Since 29 is a prime number and does not divide evenly into \( y \), the fraction \( \frac{29}{y} \) is already in its lowest terms.

Key Concepts

Adding Fractions

When we talk about adding fractions, the main goal is to combine two or more fractions into a single fraction. In order to achieve this, the fractions need to have the same denominator, which is also known as having a common denominator. In the given exercise, the fractions \( \frac{17}{y} \) and \( \frac{12}{y} \) already share the same denominator \( y \). This makes the addition process straightforward. To add fractions with the same denominator:

• Add the numerators together while keeping the denominator unchanged.
• For example, \( \frac{17}{y} + \frac{12}{y} \) becomes \( \frac{17+12}{y} \).
• You simply add the top numbers (17 and 12) and keep the bottom number (y) the same.

By following these steps, you'll learn that adding fractions isn't too daunting when you are dealing with common denominators. It simplifies the process and eliminates the need for complex calculations. Once you've added the numerators, you can then move to the next step of simplifying if needed.

Common Denominators

The concept of common denominators is crucial when working with fractions, especially when trying to add or subtract them. A common denominator is a shared multiple of the denominators of two or more fractions. Having a common denominator allows you to easily combine fractions, as you only need to operate on the numerators. In the problem \( \frac{17}{y} + \frac{12}{y} \), the common denominator is \( y \). Both fractions already have \( y \) as their denominator, which simplifies our job significantly. When fractions share a denominator, like these ones:

• You don't need to alter or find another denominator, saving you time and effort.
• You can directly add or subtract the numerators, while the denominator remains the same.

Understanding common denominators is essential because when they aren’t common, extra steps are needed to find an appropriate denominator, often involving finding the least common multiple.

Simplifying Fractions

After adding or subtracting fractions, the next step is to simplify the result whenever possible. Simplifying a fraction involves reducing it to its lowest terms. This means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number.In the context of our problem, once we've added the numerators to get \( \frac{29}{y} \):

• We check to see if there are any common factors between the numerator (29) and the denominator (y).
• Since 29 is a prime number and doesn't share any factor with \( y \) (apart from 1), the fraction is already in its simplest form.

Simplifying fractions is important because it makes the fraction easier to understand and use. It’s like cleaning up an equation to make it more presentable and functional.