Problem 85 - Elementary Algebra

Question

Simplify each algebraic expression by combinang similar terms. $$\frac{1}{2} x+\frac{2}{3} x+\frac{1}{6} x$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{4}{3}x \).

Step 1: Identify Like Terms

Notice that all the terms include the variable \( x \). Therefore, \( \frac{1}{2}x \), \( \frac{2}{3}x \), and \( \frac{1}{6}x \) can be combined since they are like terms.

Step 2: Find a Common Denominator

The fractions \( \frac{1}{2} \), \( \frac{2}{3} \), and \( \frac{1}{6} \) need a common denominator to be combined. The least common denominator of 2, 3, and 6 is 6.

Step 3: Convert Each Fraction to the Common Denominator

Convert \( \frac{1}{2} \) to \( \frac{3}{6} \), convert \( \frac{2}{3} \) to \( \frac{4}{6} \), and \( \frac{1}{6} \) remains the same.

Step 4: Add Fractions Together

Now add the fractions: \( \frac{3}{6}x + \frac{4}{6}x + \frac{1}{6}x = \frac{8}{6}x \).

Step 5: Simplify the Fraction

Simplify \( \frac{8}{6} \) by dividing both the numerator and the denominator by 2 to get \( \frac{4}{3} \). Thus, \( \frac{8}{6}x = \frac{4}{3}x \).

Key Concepts

Combining Like Terms

In algebra, you'll often run into expressions where you need to simplify by combining like terms. Like terms are portions of an expression that have the same variables raised to the same power. For instance, terms like \( \frac{1}{2}x \), \( \frac{2}{3}x \), and \( \frac{1}{6}x \) are all like terms because they each have the variable \( x \).
Combining like terms simplifies the expression, making it easier to work with. Here's how you do it:

• Identify all the terms with the same variables and exponent.
• Add or subtract them as needed, just like with regular numbers.
• Ensure any fractions are dealt with appropriately by finding common denominators first.

This process reduces complex algebraic expressions to more manageable forms.

Common Denominators

When dealing with fractions, finding a common denominator is crucial, especially when you add or subtract them. The least common denominator is the smallest number that is a common multiple of the denominators you have in your fractions.

For example, if you have fractions like \( \frac{1}{2} \), \( \frac{2}{3} \), and \( \frac{1}{6} \), the common denominator facilitates their addition. Here's how to find it and use it:

• List out the multiples of each denominator:
  • Multiples of 2: 2, 4, 6, 8...
  • Multiples of 3: 3, 6, 9, 12...
  • Multiples of 6: 6, 12, 18...
• Identify the smallest shared multiple (6 in this case).
• Convert each fraction so that its denominator becomes this common denominator.

This process allows you to efficiently add or subtract fractions without confusion.

Adding Fractions

Adding fractions is a skill that comes in handy often, especially in algebra. Once you have fractions with a common denominator, you can easily add them. Let's look at a step-by-step process:

• Convert each fraction to have this common denominator as seen in the previous section.
• Keep the common denominator and add the numerators together.
• Simplify if possible by reducing the fraction through division by any common factors.

In the example \( \frac{3}{6}x + \frac{4}{6}x + \frac{1}{6}x \), add the numerators while keeping \( 6 \) as the common denominator to get \( \frac{8}{6}x \).
Then simplify to \( \frac{4}{3}x \), making your math clearer and easier to process.

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but they include variables in the numerator, denominator, or both. Simplifying these fractions involves the same principles used in regular arithmetic fractions, including finding common denominators and combining like terms.

Here’s how to approach algebraic fractions:

• Handle the variables just like you would numeric coefficients.
• Apply operations such as addition, subtraction, multiplication, and division by using the rules of algebra alongside arithmetic.
• Always search for opportunities to simplify expressions by canceling out common factors and simplifying numerators and denominators.

Simplifying algebraic fractions reduces the complexity of a problem and can give you a deeper understanding of the relationships within the expression.