Problem 101 - Beginning and Intermediate Algebra

Question

Write each phrase as a mathematical expression using \(x\) as the variable, and simplify. Five times a number, added to the sum of the number and three

Step-by-Step Solution

Answer

6x + 3

Step 1 - Identify the Components

First, identify all the parts of the phrase that will be converted into mathematical expressions. Here, 'five times a number' and 'the sum of the number and three' are the key components.

Step 2 - Translate to Mathematical Language

Translate each identified component into a mathematical expression. 'Five times a number' translates to \(5x\) and 'the sum of the number and three' translates to \(x + 3\).

Step 3 - Combine the Expressions

Combine the expressions according to the phrase. 'Added to' indicates addition, so we combine them as \(5x + (x + 3)\).

Step 4 - Simplify the Expression

Finally, simplify the combined expression: \[5x + (x + 3) = 5x + x + 3 = 6x + 3\].

Key Concepts

mathematical translation

Mathematical translation involves converting words and phrases into algebraic expressions. In this exercise, we began with the phrase 'Five times a number, added to the sum of the number and three'. Understanding how to break down this phrase is the first step.

First, identify the key components within the phrase:

• 'Five times a number' becomes the product of 5 and a variable, represented as \(5x\).
• 'The sum of the number and three' is simply adding the variable, \(x\), and three, written as \(x + 3\).

Once these components are recognized, combining them using the provided words ('added to' indicates addition) creates the full expression. Thus, \(5x + (x + 3)\).

simplifying expressions

Simplifying expressions is a crucial step in algebra. It involves combining like terms to create a more condensed form of the expression. Here’s how we simplified the expression from the previous step:

Initially, we had: \(5x + (x + 3)\).
By removing the parentheses and combining like terms, we get:
\(5x + x + 3\).
Notice how \(5x\) and \(x\) are like terms. These terms can be added together:
\(5x + x = 6x\).
This leaves us with: \(6x + 3\).
The final simplified version of the expression is \(6x + 3\). Simplifying makes the expression easier to work with, especially in more complex equations or when solving for variables.

variable manipulation

Variable manipulation is the process of working with variables to isolate, combine, or simplify them. In this exercise, we manipulated the variable \(x\) to create and simplify the algebraic expression.

Initially, we converted parts of the phrase into expressions involving the variable \(x\). Afterward, we combined these expressions while following algebraic rules.

Throughout this process, remember that:

• Variables represent unknown values.
• Combining like terms (terms with the same variable) reduces complexity.
• Simplifying keeps expressions manageable and clear.

By carefully translating, combining, and simplifying, variable manipulation becomes a powerful tool in solving algebra problems. It helps to make confusing word problems into straightforward math equations.