Problem 36 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{5 x}{3}+\frac{2 x}{9}$$

Step-by-Step Solution

Answer

\( \frac{17x}{9} \)

Step 1: Identify the Fractions

The given expression is \( \frac{5x}{3} + \frac{2x}{9} \). We need to add these two fractions.

Step 2: Find the Least Common Denominator (LCD)

To add fractions with different denominators, we find the least common denominator. The denominators are 3 and 9. The least common multiple of 3 and 9 is 9. Thus, the LCD is 9.

Step 3: Adjust the First Fraction

Convert \( \frac{5x}{3} \) so it has the denominator 9. To do this, we multiply both the numerator and the denominator by 3, giving us \( \frac{5x \times 3}{3 \times 3} = \frac{15x}{9} \).

Step 4: Rewrite the Expression with a Common Denominator

Now rewrite the expression using the common denominator: \( \frac{15x}{9} + \frac{2x}{9} \).

Step 5: Add the Numerators

Since the denominators are the same, we add the numerators: \( 15x + 2x = 17x \). The expression becomes \( \frac{17x}{9} \).

Step 6: Simplify the Result (if possible)

Check if the fraction can be simplified. \( \frac{17x}{9} \) is already in simplest form since 17 is a prime number and does not divide 9. Thus, the fraction is fully simplified.

Key Concepts

Least Common Denominator

The least common denominator (LCD) is essential when adding or subtracting fractions with different denominators. Imagine you have apples and oranges, each grouped in different ways. To combine them, you need a common way to count them, much like our denominators in fractions.
The LCD is the smallest number that all denominators can divide into equally. For example, with denominators 3 and 9, you start by listing multiples:

• Multiples of 3: 3, 6, 9, 12...
• Multiples of 9: 9, 18, 27...

From these lists, 9 is the smallest number appearing in both, making it the least common denominator. Having common denominators simplifies the addition or subtraction of fractions because it standardizes the parts into which the whole has been divided. You basically turn them into a common language, much like turning currency into a single standard.

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but they include variables, such as x or y, in the numerator or denominator. They follow the same rules as numeric fractions when it comes to operations like addition or subtraction. Let's look at the fraction \( \frac{5x}{3} \).

• If your variable terms share the same denominator, you can directly add or subtract them, just like with numbers.
• If not, you'll need to adjust them so they do, which usually involves finding an LCD and manipulating the terms to match this standard.

For our equation \( \frac{5x}{3} + \frac{2x}{9} \), we first need to deal with the different denominators. We find the LCD to make the expressions compatible for addition. This compatibility means all fractions share the same-sized slice of pie, so to speak, allowing for them to be added together neatly.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as straightforward as possible by reducing it to its smallest equivalent form. This involves a few simple steps:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCF.
• If no GCF exists other than 1, the fraction is already in its simplest form.

For a fraction like \( \frac{17x}{9} \), we check if 17 and 9 share any factors other than 1. Since 17 is a prime number and doesn’t share factors with 9, this fraction is already simplified.
Simplifying helps in maintaining the clarity of expressions and in making calculations easier, much like cleaning your room helps you find things faster.