Problem 103 - Beginning and Intermediate Algebra

Question

Write each phrase as a mathematical expression using \(x\) as the variable, and simplify. A number multiplied by \(-7,\) subtracted from the sum of 13 and six times the number

Step-by-Step Solution

Answer

13 + 13x

Step 1: Define the phrases in terms of variables

Let the number be represented as the variable \(x\).

Step 2: Translate each part of the phrase into a mathematical expression

The phrase 'a number multiplied by \(-7\)' can be written as \(-7x\).

Step 3: Translate the sum part of the expression

The phrase 'the sum of 13 and six times the number' can be written as \(13 + 6x\).

Step 4: Write and combine the mathematical expressions

Combining the expressions, the complete phrase 'a number multiplied by \(-7\), subtracted from the sum of 13 and six times the number' translates to: \((13 + 6x) - (-7x)\).

Step 5: Simplify the expression

Perform the simplification: \(13 + 6x + 7x = 13 + 13x\).

Key Concepts

Translating phrases to expressions

Translating verbal expressions into algebraic expressions is a crucial skill in algebra. It involves interpreting words and phrases into mathematical language. Let's break down the given phrase step by step.
First, identify the variable. In this exercise, the variable is represented by \(x\), which stands for 'a number'. Next, translate individual parts of the phrase:

• 'A number multiplied by -7' becomes \(-7x\).
• 'Sum of 13 and six times the number' translates to \(13 + 6x\).

Finally, combine the parts into one expression. The phrase 'subtracted from' indicates that we need to subtract the first part from the second part, leading to the complete expression: \((13 + 6x) - (-7x)\).

Combining like terms

Combining like terms is essential for simplifying algebraic expressions. Like terms have the same variable raised to the same power. For example, \(6x\) and \(7x\) are like terms.
In our expression \((13 + 6x) - (-7x)\),\ the terms \6x\ and \-7x\ both contain the variable \(x\), making them like terms. When combining like terms, perform the arithmetic operations to simplify:

• Add \6x\ and \7x\ to get \13x\.
• Combine constants separately. Here we only have one constant: \13\.

The simplified version of the expression is \13 + 13x\.

Variables in algebra

Variables are symbols that represent numbers, and they're fundamental in algebra. They allow us to construct general formulas and solve equations.
In the exercise, \(x\) is our variable, standing for 'a number'. By using variables, we can generalize mathematical expressions and work with unknown quantities. This flexibility makes algebra a powerful tool.
A variable like \(x\) can take on any value, and its meaning is determined by its context within an expression or equation. In our case, expressions such as \6x\, \-7x\, and \((13 + 6x) - (-7x)\)\ become manageable and meaningful through the use of the variable \(x\).