Problem 30 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. the sum of a number and 6

Step-by-Step Solution

Answer

The algebraic expression for the English phrase 'the sum of a number and 6' is \(x + 6\)

Step 1: Identify the variable

In this case, the exercise has already specified that the variable to be used is \(x\). This is the entity that represents our unknown number.

Step 2: Understand the operation expressed in the problem

The phrase 'the sum of' indicates an addition operation. We will be adding together the variable (\(x\)) and a specific value (6).

Step 3: Write down the algebraic expression

Combining our variable and the operation, we can write down the algebraic expression corresponding to the English phrase. In this case, that expression is \(x + 6\).

Key Concepts

Variables in Algebra

Algebra relies on the use of variables to represent unknown quantities. Variables are symbols, typically letters such as x, y, or z, that stand in for numbers we do not yet know or are not specified. They are a fundamental part of algebraic expressions and equations that allow us to generalize problems and find solutions regardless of the specific numbers involved.

For example, if we have a scenario involving an unknown amount of money, we might let the variable x represent this quantity. If we are told that we have x dollars and someone gives us 5 more dollars, the total amount we have is represented by the expression x + 5. Without variables, it would be much harder to write down and solve this problem if we didn't know the specific amount of money to start with.

It's important to choose an appropriate variable that makes sense in the context of a problem and to be consistent in using this variable across related algebraic expressions and equations.

Translating English to Algebra

When solving algebraic problems, it is often necessary to translate words and phrases from English (or another language) into algebraic expressions. This is a critical skill in math as it allows us to bridge the gap between real-world scenarios and abstract mathematical concepts.

To translate a phrase like 'the sum of a number and 6', start by identifying the operation involved - in this case, 'sum' indicates addition. Next, determine the components of the operation. The 'number' is unspecified, so we choose a variable, often x, to represent it. The other component is the number 6. Then, we connect them using the plus sign (+), which denotes addition, to form the algebraic expression x + 6.

Tips for Translating:

• Look for keywords that indicate mathematical operations: 'sum' for addition, 'product' for multiplication, 'difference' for subtraction, and 'quotient' for division.
• Identify any specific numbers mentioned and the variables.
• Keep the order of terms in mind - in English, we might say 'a number and 6', but in algebra, the variable usually comes first.
• Remember to translate phrases like 'more than' or 'less than' correctly, as they can affect the order of the terms in the expression.

Algebraic Addition

Algebraic addition is one of the foundational operations in algebra, performed by combining numbers, variables, or both. The operation is indicated by the plus sign (+). When adding algebraic expressions, we only combine like terms, which means terms with the same variables raised to the same power.

For instance, in the algebraic expression x + 6, we are adding the variable x to the number 6. In this case, there are no like terms to combine further, so the expression remains as is. However, if we had an expression such as x + x, we could combine these like terms because they both contain the variable x and are both to the power of 1 (understood when the power is not written). This would result in 2x.

Practice with Like Terms:

• Combine 3x + 2x: The like terms 3x and 2x can be added to get 5x.
• Do not combine 3x + 2y: Here, 3x and 2y are not like terms because they have different variables.

Understanding algebraic addition is crucial when simplifying expressions and solving equations. Always look for like terms and perform any addition or simplification possible to reduce the expression to its simplest form.