Problem 37 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{19-3 \cdot 5}{6-4} $$

Step-by-Step Solution

Answer

The expression simplifies to 2.

Step 1: Simplify expressions in the numerator

Start by simplifying the expression in the numerator: \(19 - 3 \cdot 5\). First, perform the multiplication: \(3 \cdot 5 = 15\). Now substitute back into the expression to get \(19 - 15\).

Step 2: Complete the subtraction in the numerator

Continue by subtracting the numbers you obtained in the previous step: \(19 - 15 = 4\). So the numerator simplifies to 4.

Step 3: Simplify expressions in the denominator

Next, simplify the expression in the denominator: \(6 - 4\). Simply subtract to get \(6 - 4 = 2\). So the denominator simplifies to 2.

Step 4: Divide the simplified numerator by the simplified denominator

Now, divide the simplified numerator by the simplified denominator: \(\frac{4}{2}\). Perform the division to get \(2\). Therefore, the original expression simplifies to \(2\).

Key Concepts

Understanding the Numerator and Denominator

In an algebraic expression such as a fraction, the top part is known as the numerator and the bottom part is known as the denominator. These components are crucial when simplifying expressions.

• Numerator: This is the part above the fraction line and represents the number of parts we have. In our exercise, the numerator was the expression \(19 - 3 \cdot 5\).
• Denominator: This is the part below the fraction line and indicates the number of equal parts the whole is divided into. For the given problem, the denominator was \(6 - 4\).

To simplify a fraction, you first simplify both the numerator and denominator independently. This means performing any operations within each part before you move on to dividing them. Understanding how each part of a fraction behaves allows you to approach simplifying with clarity and purpose.

The Order of Operations in Simplification

The order in which we perform operations in a mathematical expression can greatly impact the final result. In algebra, we follow a specific sequence often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our exercise, the expression within the numerator required us to perform multiplication before subtraction. Even though \(19 - 3 \cdot 5\) might seem straightforward, according to the order of operations, multiplication comes before subtraction. Thus, \(3 \cdot 5 = 15\) needs to be calculated first, leading to the simplified expression \(19 - 15 = 4\). Similarly, the denominator was simplified by directly performing the subtraction \(6 - 4 = 2\). Recognizing and adhering to these rules ensures the accuracy of your simplification.

Simplifying Using Basic Arithmetic Operations

Basic arithmetic operations are the foundation of simplifying expressions. These operations include addition, subtraction, multiplication, and division.

• Addition and Subtraction: These operations combine or take apart values. In the numerator, we subtracted 15 from 19 to yield 4.
• Multiplication: This operation is used to calculate repeated addition quickly. In this example, multiplying 3 by 5 provided the value of 15, which was necessary for the subtraction that followed.
• Division: This operation divvies up a number into equal parts. The final step of our simplification was dividing 4 by 2, resulting in 2.

By fully grasping how to apply these operations individually and within larger expressions, you enhance your mathematical fluency and are better prepared to tackle even more complex algebraic problems.