Problem 102 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

7x + 6

Step 1: Identify the Number

Identify the unknown number in the problem. Let's use the variable x to represent the unknown number.

Step 2: Translate Each Part of the Phrase

The phrase 'Six times a number' translates to the mathematical expression 6x. The phrase 'the sum of the number and six' translates to the expression (x + 6).

Step 3: Combine the Expressions

The phrase 'added to' suggests adding these two expressions together. Thus, the complete expression is 6x + (x + 6).

Step 4: Simplify the Expression

Combine like terms in the expression 6x + (x + 6) to get the simplified expression 7x + 6.

Key Concepts

translating phrases to algebra

Understanding how to translate phrases into algebraic expressions is the first step in solving algebra problems. For example, the phrase 'six times a number' can be translated to the algebraic expression \(6x\), where \(x\) represents the unknown number. Similarly, the phrase 'the sum of the number and six' translates to \((x + 6)\).

To translate English phrases into algebra, look for keywords and their mathematical equivalents:

• 'Times' or 'multiplied by' often translates to multiplication (e.g., 'six times \(x\)' is \(6x\)).
• 'Sum' translates to addition (e.g., 'sum of \(x\) and 6' is \(x + 6\)).
• 'Difference' translates to subtraction.
• 'Product' refers to multiplication.
• 'Quotient' refers to division.

Mastering this step is crucial as it lays the foundation for the next steps.

combining like terms

Combining like terms is an essential skill in algebra that simplifies expressions into a more manageable form. Like terms are terms that have the same variable raised to the same power. For instance, in our initial expression \(6x + (x + 6)\), the terms \(6x\) and \(x\) are like terms because they both contain the variable \(x\).

Here's how you combine them:

• Identify like terms. For example, \(6x\) and \(x\) are like terms.
• Combine them by adding their coefficients. So, \(6x + x\) becomes \(7x\).

This gives us the simplified expression \(7x + 6\). Combining like terms makes it easier to solve or manipulate algebraic expressions.

simplification

Simplification is the process of making an algebraic expression as straightforward as possible. After translating the original phrase to an algebraic expression and combining like terms, the final step is to simplify. This means combining all reducible parts of the expression.

Let's look again at our expression: Initially, we had \(6x + (x + 6)\). After combining like terms, we get \(7x + 6\).

Tips for simplification:

• Combine like terms.
• Perform any arithmetic operations that are possible.

Simplifying helps make the expression clearer and easier to work with. In this particular problem, our simplified expression is \(7x + 6\), fully reduced and ready for any further use.