Problem 11 - Elementary and Intermediate Algebra

Question

Classify each of the following as either an expression or an equation. $$ r(t+7)+5 $$

Step-by-Step Solution

Answer

Expression

Step 1: Identify Terms and Operations

Observe the given mathematical structure, which is \(r(t+7)+5\), and note that it consists of variables (r and t), a constant (7 and 5), and operations (addition and multiplication).

Step 2: Check for an Equals Sign

An equation will always have an equals sign (\(=\)) indicating that two expressions are the same. Check if \(r(t+7)+5\) includes an equals sign.

Step 3: Classification

Since \(r(t+7)+5\) does not contain an equals sign, it is classified as an expression.

Key Concepts

Understanding Expressions

In algebra, an expression is a combination of numbers, variables (like r and t), and mathematical operations (like addition and multiplication). Expressions are used to represent a value and they do not include an equals sign (=).For example, in the expression r(t+7)+5, we see:

• Variables: r, t
• Constants: 7, 5
• Operations: Addition, Multiplication

Each part of an expression is called a term. Here, r(t+7) and 5 are terms combined by the operation of addition.Expressions can be simplified or expanded, but they cannot be solved because they do not state a relationship of equivalency (like an equation does).Understanding expressions is fundamental to grasping more complex algebraic structures.

What Makes an Equation?

Equations, unlike expressions, contain an equals sign (=).They show that two expressions are equal. For example, an equation can look like this:\[ 2x + 3 = 11 \]This equation tells us that when we multiply 2 by some number x and add 3, the result is 11. Equations are used to determine the value of unknown variables by finding the values that make both sides of the equation equal.To solve an equation, we perform operations that maintain the balance, isolating the variable on one side of the equals sign. This process is often referred to as 'solving for x' or any other variable.Equations are essential for solving problems and modeling real-world situations in mathematics.

Basic Mathematical Operations

Mathematical operations are the basic building blocks used to manipulate numbers and variables. The fundamental operations include:

• Addition (+): Combining two numbers or quantities to get a sum.
• Subtraction (−): Finding the difference between two numbers or quantities.
• Multiplication (×): Calculating the total of one number added to itself a certain number of times.
• Division (÷): Determining how many times one number is contained within another.

In the expression r(t+7)+5, we see both addition and multiplication:

• Addition is used to combine (t+7) and 5.
• Multiplication is used between r and (t+7).

Understanding these operations is essential because they form the basis of simplifying and evaluating expressions and solving equations. Getting comfortable with these operations enhances problem-solving skills in algebra and beyond.