Problem 60 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{3}{s-8}+t $$

Step-by-Step Solution

Answer

The expression \( \frac{3}{s-8} + t \) cannot be simplified further without additional information.

Step 1: Identify the expression type

The expression given is \( \frac{3}{s-8} + t \), which consists of a fraction with a variable in the denominator and an addition with another variable.

Step 2: Look for a common denominator

Since the terms involve different variables and are not fractions with common denominators, we can't combine them into a simpler single fraction without additional context or substitution for \(s\) or \(t\).

Step 3: Simplify components separately

The fraction \( \frac{3}{s-8} \) is as simplified as possible as it is a proper fraction. Without substitution or additional constraints on \(t\), the expression cannot be simplified further.

Key Concepts

Fractions in Algebra

Algebraic fractions are similar to numerical fractions, but they contain variables as part of the numerator, denominator, or both. When working with fractions in algebra, remember the same rules you use for regular numbers apply. The challenge occurs when variables are involved.

For instance, in the expression \( \frac{3}{s-8} + t \), you have a fraction where the denominator has a variable \(s\). Such algebraic fractions often need careful handling, especially when you're required to perform operations like addition or subtraction. It's important to look for opportunities to find common denominators when working with multiple algebraic fractions. Though in this specific case, the term \(t\) isn't part of a fraction, so combining both terms into a single fraction expression without extra context is not straightforward.

Simplifying Expressions

Simplifying expressions in algebra means reducing them to their simplest form while still representing the original values they stand for. You often want to make expressions more manageable, especially when they involve fractions or variables.

In the problem \( \frac{3}{s-8} + t \), simplifying directly isn't feasible without extra steps or information about the variables. The term \( \frac{3}{s-8} \) is already in its simplest form as a fraction since there's no common factor between 3 and \(s-8\). Moreover, the addition with \(t\) remains as it is unless there’s further information about the relationship between \(s\) and \(t\), such as through substitutions or equations. Simplification didn't apply in this case, but understanding when an expression is 'as simple as it can be' is vital in algebra.

Variables in Denominators

Variables in the denominator can make algebraic expressions more complex because they introduce the possibility of undefined values. If the variable in the denominator takes a value that makes the denominator zero, the expression becomes invalid.

Take \( \frac{3}{s-8} \) as an example. Here, the denominator has the variable \(s\). You must ensure \(s eq 8\) because if \(s = 8\), the expression \( \frac{3}{s-8} \) becomes undefined (division by zero). It's also important to identify and specify such variable restrictions when interpreting or simplifying algebraic expressions. Therefore, when handling fractions with variables in the denominator, always consider the conditions under which the expression maintains its value.