Problem 18 - Algebra 2

Question

Write an algebraic expression to represent each verbal expression. seven more than the product of a number and 10

Step-by-Step Solution

Answer

The algebraic expression is \( 10x + 7 \).

Step 1: Identify Key Components

The verbal expression consists of: 1) a product of a number and 10, and 2) having seven more than this product. The 'number' is an unknown quantity, which we often represent as a variable in algebra, typically \( x \).

Step 2: Represent the Product

The product of a number and 10 can be expressed algebraically as \( 10x \), where \( x \) is the unknown number.

Step 3: Add Seven to the Product

The phrase "seven more than" indicates that we need to add 7 to the previous expression. Therefore, we take the product \( 10x \) and add 7 to it. This gives the expression: \( 10x + 7 \).

Key Concepts

Algebraic Variables

In algebra, variables are symbols used to represent unknown values. They act as placeholders for real numbers that can change or vary. Typically, letters such as \( x \), \( y \), or \( z \) are used as variables. In the exercise, we used \( x \) to represent the unknown number. Using variables allows us to construct expressions and equations to solve problems more effectively.
Algebraic variables can:

• Help generalize mathematical concepts
• Facilitate the solving of equations
• Allow for the representation of relationships between quantities

Variables are a fundamental concept in algebra, making abstract representations of problems easier to manage and understand. Understanding how to use variables effectively is key to mastering algebra and developing problem-solving skills.

Verbal Expressions in Algebra

Verbal expressions in algebra translate everyday language into mathematical expressions. This is an essential skill because daily problems often appear in sentence form and need to be converted into mathematical terms to be analyzed or solved.
Consider the expression "seven more than the product of a number and 10." In this verbal expression, we need to identify components:

• "Product of a number and 10" means multiplying a number (our variable \( x \)) by 10, resulting in \( 10x \).
• "Seven more than" indicates addition, so we add 7 to the product, resulting in \( 10x + 7 \).

Understanding verbal expressions helps bridge the gap between language and mathematics, enabling the translation of word problems into algebraic form. This skill is particularly useful in real-world applications of algebra.

Writing Expressions in Algebra

Writing expressions in algebra involves converting verbal descriptions into mathematical statements. This process requires identifying the mathematical operations described by words and choosing appropriate symbols and variables to reflect these operations.
For the phrase "seven more than the product of a number and 10":

• "A number" is unknown, typically represented by \( x \).
• "Product of a number and 10" means multiplying, written as \( 10x \).
• "Seven more than" requires adding 7 to the product, forming \( 10x + 7 \).

By breaking down verbal expressions into parts, we can incrementally write accurate algebraic expressions. Mastery of this skill is crucial for solving complex problems and for making sense of mathematical language in textbooks and real-life situations.