Problem 60 - Elementary Algebra

Question

Use algebraic notation to write "eleven minus three times a number is five."

Step-by-Step Solution

Answer

Question: Translate the statement "eleven minus three times a number is five" into an algebraic equation and identify the variable used for the unknown number. Answer: The algebraic equation is 11 - 3 * x = 5, where the variable x represents the unknown number.

Step 1: Identify the unknown number

Let's represent the unknown number by the variable x.

Step 2: Translate the statement into an algebraic expression

The statement "eleven minus three times a number" can be translated to the algebraic expression: 11 - 3 * x

Step 3: Write the complete algebraic equation

The full statement says "eleven minus three times a number is five." Using the algebraic expression from Step 2, we can write the complete equation as: 11 - 3 * x = 5

Key Concepts

Variables

In algebra, a variable is a simple symbol, typically a letter, that stands in for an unknown quantity. Variables are the building blocks of algebraic expressions and equations. In the context of our original exercise, we used the variable \(x\) to represent the "unknown number."
This choice allows us to work with a general number whose value can vary, thus the name 'variable'.

• Variables are essential because they allow us to write general rules and formulas that we can apply to many situations.
• By using variables, we can represent numbers without specifying their exact value right away.
• They are placeholders and can represent different values in different situations until we have enough information to solve the equation.

Understanding how to use variables is crucial when moving from arithmetic to algebra. They enable us to express a wide range of mathematical concepts and solve problems systematically.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. It's a crucial part of algebra that allows us to express complex ideas concisely.
In our exercise, we translated the phrase "eleven minus three times a number" into an algebraic expression: \(11 - 3 \cdot x\).

• An algebraic expression doesn't have an equal sign; it's not a complete statement until it's part of an equation.
• They can represent real-world situations. For instance, "3 times the cost of a ticket" could be represented by \(3 \cdot c\), where \(c\) is the cost.
• They can be simplified by combining like terms or using arithmetic operations.

Working with algebraic expressions involves understanding and applying basic arithmetic, as well as recognizing patterns and structures within expressions.

Equations

Equations are mathematical statements that assert the equality of two expressions. They are powerful tools for finding unknown values and solving practical problems.
In our exercise, we took the expression \(11 - 3 \cdot x\) and equated it to \(5\) to solve the statement: "eleven minus three times a number is five." This gave us the equation \(11 - 3 \cdot x = 5\).

• Equations contain two expressions separated by an equal sign, which shows that they are balanced.
• Solving an equation involves finding the value of the variable that makes the equation true.
• They can model real-life situations, such as calculating distances, speeds, and other quantities.

Learning to work with equations is essential because it allows us to systematically determine values and understand relationships between quantities. Equations form the backbone of algebra and play a pivotal role in advanced mathematics and science.