Problem 16 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{4 t-1}{7}+\frac{8 r-5}{7}$$

Step-by-Step Solution

Answer

\(\frac{4t + 8r - 6}{7}\)

Step 1: Identify the operation

The problem asks us to either add or subtract the two given expressions. Here, we are given two fractions to add: \(\frac{4t-1}{7}\) and \(\frac{8r-5}{7}\). Both fractions have the same denominator, which simplifies the process to add them.

Step 2: Add the numerators

Since the denominators are equal, we can directly add the numerators of both fractions while keeping the denominator the same: \(\frac{(4t-1) + (8r-5)}{7}\).

Step 3: Simplify the numerator

Simplify the expression in the numerator by combining like terms: \((4t - 1) + (8r - 5) = 4t + 8r - 6\).

Step 4: Express the fraction in simplest form

The resulting expression is \(\frac{4t + 8r - 6}{7}\). There are no common factors in the numerator that can be further simplified, so this is already in its simplest form.

Key Concepts

Addition of Fractions

Adding fractions might seem tricky at first, but the key is to focus on the numerators if the denominators are already the same. When two fractions share a common denominator like in this problem \(\frac{4t-1}{7}\) and \(\frac{8r-5}{7}\), the process becomes straightforward:

• Keep the denominator unchanged. In this case, our denominator is 7 for both fractions.
• Simply add the numerators together. For our fractions, that means adding \((4t-1)\) and \((8r-5)\).

Conveniently, since the denominators match, no additional adjustments are necessary to perform the addition. Always remember: only numerators are added when denominators are the same.

Simplifying Fractions

Once you've combined the numerators of fractional expressions, the next step is to simplify them. Simplifying isn't just about getting a tidy answer; it's about making the fraction as concise as possible.
You'll want to:

• Combine like terms. In our expression, \((4t-1) + (8r-5)\), you identify terms that can be combined: \(4t\) and \(8r\), followed by the constants \(-1\) and \(-5\).
• Perform the arithmetic: \(4t + 8r - 6\). Adding or subtracting similar terms simplifies the expression effectively.

After combining and simplifying the terms, you should check if there are any common factors among the terms in the numerator that could further reduce the fraction over the shared denominator.

Common Denominators

Having common denominators makes adding or subtracting fractions much easier. Without a shared denominator, you would have to find a common one, usually the smallest possible value.

• Common means identical – the fractions here already have 7 as the denominator.
• If denominators differ, you would need to calculate the least common multiple (LCM) of the denominators as the new shared base.

This problem is simpler since both fractions already have a denominator of 7, allowing direct addition of numerators without further adjustment. Recognizing and working with common denominators is crucial for dealing with more complex algebraic fractions efficiently.