Problem 65 - Prealgebra

Question

Write the mathematical expressions that are equivalent to each of the following English phrases. The sum of twice a number and 5

Step-by-Step Solution

Answer

The expression is \( 2x + 5 \).

Step 1: Identify the Variable

First, we need to define a variable to represent the unknown number. Let's use the letter \( x \) to represent this number.

Step 2: Translate "Twice a Number"

Twice a number means 2 times the number. Therefore, the expression for "twice a number" is \( 2x \).

Step 3: Translate "Sum of"

The phrase "sum of" indicates an addition operation. We will add the results of the other expressions to get the sum.

Step 4: Incorporate the Constant

The constant mentioned in the phrase is 5, which needs to be added to the expression from Step 2.

Step 5: Combine the Expressions

Finally, combine the expressions for twice the number and the constant 5 using the addition operation: \( 2x + 5 \).

Key Concepts

Mathematical Expressions

In the world of mathematics, expressions are like sentences in our daily language. Instead of words, they use numbers, symbols, and operations. These combinations can show numbers, operations, and variables. Variables, often represented by letters like \( x \), stand in for unknown values. Expressions might not always have an "equal" sign, which is what makes them different from equations.

• An example of a simple expression might be \( 5 + 3 \).
• Once variables enter the mix, the expression evolves, such as \( 2x + 5 \).

Understanding the basic structure of mathematical expressions is important for diving into algebra. They allow you to describe and mathetically model real-world situations.

Translating Phrases to Expressions

Often, mathematical problems begin with an English phrase that needs to be translated into a mathematical expression. This is crucial for turning word problems into solvable math problems. Let's break down how to approach this translation process:- **Identify Key Terms:** Start by picking out words that indicate mathematical operations, such as "sum" for addition or "product" for multiplication. The phrase "the sum of" tells us to use addition.- **Define the Variable:** Choose a variable to represent unknown numbers in the phrase. For example, "a number" can be symbolized by \( x \).- **Translate Each Part:** Convert each segment of the phrase individually. For instance, "twice a number" becomes \( 2x \) since it means two times \( x \).Once each part is translated, you will be able to combine them, forming a complete expression such as \( 2x + 5 \). Mastering this skill is essential, especially as problems become more complex in algebra.

Beginner Algebra Process

Algebra might seem challenging at first, but breaking it down into a series of simple steps can make it more approachable.1. **Identify and Assign Variables:** Start by recognizing what the unknown components are and assign them variables to work with.2. **Break Down the Problem:** Analyzing each part of the algebraic phrase is key. For instance, understanding that "twice a number" is a multiplication task.3. **Construct the Expression:** Piece together your findings in a coherent expression. Ensure you reflect the operation implied by the phrase. In the example, that would be \( 2x + 5 \).4. **Revisit and Confirm:** Go over your constructed expression to ensure all parts of the original phrase have been captured correctly.Approaching algebra with these basic steps allows you to simplify potentially daunting problems into manageable tasks. With practice, these processes will become second nature.