Problem 38 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{14-2 \cdot 3}{12-8} $$

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Simplify the Numerator

First, resolve the expression in the numerator: \(14 - 2 \cdot 3\). Calculate \(2 \cdot 3\) which is \(6\), then perform the subtraction: \(14 - 6 = 8\). So, the numerator simplifies to \(8\).

Step 2: Simplify the Denominator

Now, simplify the expression in the denominator: \(12 - 8\). Perform the subtraction: \(12 - 8 = 4\). So, the denominator simplifies to \(4\).

Step 3: Divide the Simplified Numerator by the Denominator

Now, divide the simplified numerator by the simplified denominator: \(\frac{8}{4}\). This equals \(2\).

Key Concepts

Simplification Process

Simplifying algebraic expressions involves reducing an expression into its simplest form. This mainly includes reducing fractions, eliminating unnecessary terms, and applying arithmetic operations effectively. The main goal of simplification is to make complex expressions easier to understand and solve. Here's how you can approach it:

• Identify parts of the expression: Look for operations that can be simplified such as addition, subtraction, multiplication, and division.
• Simplify step-by-step: Focus on one part of the expression at a time, like the numerator or the denominator.
• Check your work: Make sure that you correctly follow mathematical conventions and verify your solutions.

By systematically simplifying each part of an expression, we can make the final calculation much more manageable.

Numerator and Denominator

Understanding the roles of the numerator and denominator is vital in working with fractions and algebraic expressions. In any fraction, the numerator is the expression or number above the fraction line, while the denominator is below. Together, they represent division.

Numerators and denominators can be composed of singular numbers, variables, or more intricate expressions. In our exercise, the fraction was \(\frac{14-2 \cdot 3}{12-8}\). Let's break it down:

• Numerator: Initially \(14 - 2 \cdot 3\), simplifies to \(8\).
• Denominator: Initially \(12 - 8\), simplifies to \(4\).

The division of the numerator by the denominator gives us our final simplified value. A deep understanding of how to manipulate these parts leads to correctly solving such expressions.

Order of Operations

Order of operations is a fundamental rule that defines the correct sequence to evaluate a mathematical expression. This is crucial when expressions involve multiple operations. The generally accepted sequence is bracketed as PEMDAS/BODMAS, which stands for:

• P/B: Parentheses/Brackets
• E/O: Exponents/Orders (powers and roots)
• MD: Multiplication and Division (from left to right)
• AS: Addition and Subtraction (from left to right)

In our example expression \(\frac{14-2 \cdot 3}{12-8}\), we start with multiplication in the numerator because it's prioritized over subtraction. This means solving \(2 \cdot 3\) before performing other operations. By following the order of operations meticulously, we ensure accurate and consistent results across various expressions.