Problem 15 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{3 t-1}{4}+\frac{2 r+3}{4}$$

Step-by-Step Solution

Answer

\(\frac{3t + 2r + 2}{4}\)

Step 1: Identify Common Denominator

The denominators of both fractions are the same: 4. Since they already share a common denominator, we can proceed to add the numerators.

Step 2: Combine the Numerators

Add the numerators of the two fractions together: \((3t - 1) + (2r + 3)\). This results in a new numerator: \(3t + 2r + 2\).

Step 3: Simplify the Expression

With the new numerator, the expression becomes \(\frac{3t + 2r + 2}{4}\). This is the expression in its simplest form since there are no like terms to combine further.

Key Concepts

Adding Fractions

Adding fractions might seem challenging at first, but with a few simple steps you can master it. The key is to focus on the numerators and denominators. Here, both fractions have a denominator of 4, which means they share the same base or common denominator.

• Always begin with checking whether the denominators are the same. If they are not, you would need to find a common denominator before you start adding the fractions together.
• Once the denominators are the same, you can directly add the numerators. This avoids potential errors and makes adding fractions straightforward.
• For our example, since the denominators are both 4, we added \((3t - 1)\) and \((2r + 3)\) to get \(3t + 2r + 2\).

Recognizing when to apply these steps can guide you through solving fraction problems easily. Be sure to simplify your final answer whenever possible to achieve the neatest solution.

Common Denominator

A common denominator is crucial when adding or subtracting fractions. This is because fractions represent parts of a whole, so ensuring the "size" of these parts is the same is important.

• For our exercise, both fractions already had a common denominator of 4, which simplified the addition process.
• If the denominators were different, we would find the least common denominator (LCD) and adjust each fraction accordingly before adding.
• The LCD is the smallest number that both original denominators divide evenly into. Often, it's a multiple of the original denominators.

Working with a common denominator allows the fractions to be added directly, focusing on combining the numerators over this shared value. This ensures the resulting fraction accurately represents the sum of the parts.

Simplifying Expressions

Simplifying expressions is a critical final step to ensure clarity and precision in your solutions. This means condensing your answer into its simplest form.

• After combining the numerators, as we did to achieve \(3t + 2r + 2\), we reviewed the expression for any common terms or factors to condense.
• If there were common factors in both the numerator and denominator, it would be important to factor these out where possible.
• In our case, there were no such terms, meaning the expression \(\frac{3t + 2r + 2}{4}\) was already as simplified as it could be.

Simplifying not only makes an expression easier to read, but it might reveal other equivalent forms that can be further utilized in mathematical operations. It’s always worth taking the time to ensure your solution is as straightforward as possible.