Problem 68 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression and simplify if possible. Let \(x\) represent the unknown number. Eight more than triple a number

Step-by-Step Solution

Answer

The expression is 3x + 8.

Step 1: Understand the Phrase

Interpret the phrase 'Eight more than triple a number'. Here, 'triple a number' means thrice or three times a number, which can be represented as \(3x\) where \(x\) is the unknown number.

Step 2: Write the Expression

The phrase 'eight more than triple a number' indicates that we need to add 8 to three times the number. This means the expression is \(3x + 8\).

Step 3: Simplify the Expression

Check if the expression \(3x + 8\) can be further simplified. Since there are no like terms to combine, this expression is already in its simplest form.

Key Concepts

Interpreting Algebraic Phrases

Understanding algebraic phrases involves translating words into mathematical language. Let's break down the phrase "Eight more than triple a number." Here, "triple" refers to multiplying by three, and "a number" signifies our variable, often called the unknown number, represented by the variable \(x\). By combining these parts, "triple a number" translates to the expression \(3x\).
The term "eight more than" suggests an addition of 8 to the previous amount. Therefore, we interpret this phrase as the sum of \(3x\) and 8, which together forms the algebraic expression \(3x + 8\).
Translating phrases to expressions requires attention to detail. It's about identifying keywords like "triple" or "more than" and knowing what they translate into mathematically. This skill is key in algebra, helping understand and solve a wide variety of problems.

Simplification of Expressions

Simplifying algebraic expressions is like tidying up a room. You want everything in its place. An expression is in simplest form when there are no like terms to combine or operations to perform. Consider the expression \(3x + 8\).
Since \(3x\) and 8 involve different terms (\(3x\) includes the variable while 8 is a constant), they cannot be combined. The expression is already organized and cannot be further simplified.
Simplification involves looking for like terms—terms with the same variable and exponent. In this context, 8 (a constant) and \(3x\) (a term with a variable) do not match, hence simplification is complete. Understanding this step helps in evaluating and solving equations efficiently.

Variables in Algebra

Variables are fundamental components in algebra. They stand in for unknown numbers that we want to find or understand within equations or expressions. Typically denoted by letters such as \(x\), \(y\), or \(z\), variables are placeholders that help us model real-world situations mathematically.
In our example, the variable \(x\) represents the unknown number. This concept allows us to generalize problems and create equations that describe relationships between different quantities.
Working with variables helps develop a deeper understanding of mathematical concepts. It allows flexibility in solving for different scenarios by changing the values of these unknowns. Understanding how to manipulate variables is crucial in advancing to more complex algebraic processes, like solving equations and inequalities.