Problem 36 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. five times a number, decreased by 3

Step-by-Step Solution

Answer

The English phrase 'five times a number, decreased by 3' translates into the algebraic expression \(5x - 3\).

Step 1: Identify the number and the operations

From the phrase 'five times a number, decreased by 3', it is known that the number is represented by variable \(x\) and two operations need to be performed on it. First, the number needs to be multiplied by five, and then 3 needs to be subtracted from it.

Step 2: Convert 'five times a number' into algebraic expression

The phrase 'five times a number' means the number is multiplied by 5. As \(x\) represents the number, it can be written as \(5x\).

Step 3: Convert 'decreased by 3' into algebraic expression

The phrase 'decreased by 3' means subtracting 3 from the result of the previous operation. So, it is written as \(-3\).

Step 4: Combine both expressions

The full phrase 'five times a number, decreased by 3' is translated into mathematical expression by combining both expressions obtained in steps 2 and 3. 'Decreased by 3' is a subtraction operation, so the final expression becomes \(5x - 3\).

Key Concepts

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. This allows us to generalize arithmetic operations and solve problems using variables instead of specific numbers.
One of the fundamental purposes of algebra is to find unknown values and make predictions. We use equations to describe relationships between quantities and to solve them, we apply rules of algebraic manipulation.
Key elements of algebra include:

• Expressions: Combinations of numbers, variables, and operations (like addition or multiplication). Example: \(2x + 5\).
• Equations: Statements showing the equality of two expressions. Example: \(3x + 2 = 11\).
• Inequalities: Mathematical statements indicating one expression is larger or smaller than another. Example: \(x > 5\).

Understanding algebra helps in solving real-world problems ranging from finance to engineering by modeling relationships between variables.

Variables in Algebra

In algebra, a variable is a symbol used to represent a number in expressions and equations. Variables allow us to create general solutions to problems, which makes them incredibly versatile and powerful.
In our exercise, the letter \(x\) is used as a variable to represent a number. The flexibility of using a variable means we can solve for different values depending on the situation or problem context.
Here are some important aspects of variables in algebra:

• Representation: Common variables include letters like \(x\), \(y\), and \(z\). They stand in place for unknown values.
• Constants versus Variables: Constants are fixed values (like 5 in our problem), while variables can change.
• Coefficient: A number multiplied by a variable. In our expression \(5x\), 5 is the coefficient.

Being comfortable with variables makes working with complex problems simpler, as they help translate word problems into mathematical expressions.

Mathematical Operations

Mathematical operations in algebra involve basic arithmetic functions that are used to manipulate expressions and solve equations. These operations include addition, subtraction, multiplication, and division.
Let's explore how they apply to our exercise step by step:

• Multiplication: The phrase 'five times a number' indicates a multiplication operation. The number (represented by \(x\)) is multiplied by 5, giving us \(5x\).
• Subtraction: The phrase 'decreased by 3' denotes subtraction. We start with \(5x\) and subtract 3 from it to get the final expression \(5x - 3\).

Understanding these basic operations is crucial as they form the foundation for more advanced algebraic manipulations. They allow us to transition from word-based problems to algebraic expressions neatly, making problem-solving efficient and systematic.