Problem 77 - Algebra 1: Concepts and Skills

Question

NUMERICAL EXPRESSIONS Evaluate the expression. Then simplify the answer. (Lesson 1.3) $$ 4 \cdot 2-5 $$

Step-by-Step Solution

Answer

The simplified answer is 3

Step 1: Perform Multiplication

Firstly perform the multiplication operation. Multiply 4 by 2 to get 8.

Step 2: Perform Subtraction

Subtract 5 from the result of the multiplication, which is 8, to get the final answer. Calculation is: 8 - 5.

Key Concepts

Order of Operations

When evaluating numerical expressions, it's essential to follow a standardized approach to ensure the correct result. This approach is guided by the order of operations, a fundamental concept in algebra that dictates the sequence to calculate expressions involving more than one operation.

This set of rules is often remembered through the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Keep in mind that multiplication and division are of equal precedence; the same goes for addition and subtraction. Always address them in the order they appear from left to right.

For example, in evaluating the expression \( 4 \cdot 2-5 \), we follow these rules. The expression doesn't contain parentheses, exponents, or division, so we move straight to multiplication, and then perform subtraction as shown:

• Multiplication: \( 4 \cdot 2 = 8\)
• Subtraction: \(8 - 5 = 3\)

Thus, following PEMDAS ensures we correctly interpret every numerical expression.

Simplifying Expressions

Simplifying expressions is about breaking complex calculations into simpler, more manageable steps. This not only makes it easier to understand but also minimizes the risk of making mistakes. Simplification may involve combining like terms, reducing fractions, or applying algebraic identities.

In the case of our expression \( 4 \cdot 2-5 \), there isn't much to simplify since it's relatively straightforward with just two operations. However, the process represents a microcosm of simplifying more complex expressions. You focus on parts of the equation you can handle, simplify them, and then continue until the entire expression is as simple as possible.

Here's how the simplification process can be outlined:

• Identify operations to be performed.
• Carry out the operations step by step, starting with the highest precedence (in this case, multiplication).
• Continue to the next operation (subtraction), simplifying until the expression cannot be reduced any further.

Basic Algebra Operations

Basic algebra operations are the building blocks for tackling algebraic problems and include addition, subtraction, multiplication, and division. These operations help us manipulate and solve equations or inequalities by balancing or simplifying expressions.

In the expression \( 4 \cdot 2-5 \), we use two basic operations: multiplication and subtraction. The sequence of these operations is essential in obtaining the correct result:

• Multiplication (\( \cdot \)) is used to combine numbers through repeated addition.
• Subtraction (\(-\)) is used to find the difference between numbers.

By mastering basic algebra operations and understanding how they interact based on the order of operations, you can address more complex arithmetic and algebraic tasks with confidence.