Problem 88 - Introductory Algebra for College Students

Question

Let x represent the number and write the phrase as an algebraic expression. Eight times the sum of a number and 14

Step-by-Step Solution

Answer

The algebraic expression for the phrase 'Eight times the sum of a number and 14' is \(8 (x + 14)\)

Step 1 - Identify the Variables

Recognize that 'a number' is meant to be represented by the variable x.

Step 2 - Translate 'the sum of a number and 14'

The phrase 'the sum of a number and 14' suggests the adding of our variable and 14. The mathematical operation for the sum is addition, hence this part of the phrase becomes \(x + 14\). This results in an addition term.

Step 3 - Translate 'eight times'

The phrase 'eight times' suggests multiplying the result of the sum from step 2 by 8. Hence, we surround the sum with parenthesis and place it next to the 8, resulting in \(8 (x + 14)\). The multiplication statement is represented using parentheses as they replace the multiplication sign in algebra.

Key Concepts

Variables in Algebra

In the world of algebra, variables are like placeholders. They represent unknown numbers or quantities that we are trying to find or work with. In our exercise, the phrase 'a number' is the unknown quantity and is represented by the variable \(x\). This is because algebra uses letters to stand in for numbers we don't know yet. These variables can help us write mathematical expressions and solve equations.
Using variables makes it easier to understand problems and the relationships between different quantities. As you learn more about algebra, you'll see that variables can represent simple unknowns or more complex expressions. But remember, they are just symbols that help us record information and solve problems.

Algebraic Translation

Translating words into algebraic expressions is a crucial skill in algebra. It involves converting a written phrase into a mathematical expression using variables and operators.
Let's dissect the process step by step from the original exercise. When the problem mentioned 'the sum of a number and 14', we needed to translate it into an expression. Here, 'a number' is represented by \(x\), and 'sum' signifies addition. Hence, to express 'the sum of a number and 14,' we write \(x + 14\). Translating verbal phrases into math helps us see the mathematical operations clearly so we can work with them efficiently.
Understanding algebraic translation will aid students in solving different types of problems, as it makes the connection between real-world situations and mathematical models.

Mathematical Operations

The phrase 'eight times the sum of a number and 14' involves understanding the operation of multiplication in this context. Mathematical operations such as addition, subtraction, multiplication, and division are the tools we use to work with numbers and expressions.
In our exercise, 'eight times' implies multiplication, and it means we need to multiply the expression \(x + 14\) by 8. This results in the expression \(8(x + 14)\). Here, using parentheses helps group terms together and indicates multiplication.
Knowing how to correctly apply these operations allows us to manipulate and solve algebraic expressions. It's important to recognize which operations to use from the context given and how they affect each part of the expression.