Problem 24 - Algebra A Combined Function

Question

Simplify each expression. $$ 12 \cdot 5-3 \cdot 6 $$

Step-by-Step Solution

Answer

The simplified expression is 42.

Step 1: Apply the Multiplication

Start by performing the multiplication operations indicated in the expression. Compute each multiplication separately.- First, calculate \( 12 \cdot 5 \): \[ 12 \cdot 5 = 60 \]- Then, calculate \( 3 \cdot 6 \): \[ 3 \cdot 6 = 18 \]

Step 2: Perform the Subtraction

Now, substitute the results of your multiplications back into the expression and perform the subtraction:- Replace \( 12 \cdot 5 \) with 60 and \( 3 \cdot 6 \) with 18 in the original expression: \[ 60 - 18 \]- Finally, perform the subtraction: \[ 60 - 18 = 42 \]

Key Concepts

Multiplication

In mathematics, multiplication is one of the four fundamental arithmetic operations. It involves combining groups of equal size. For instance, when multiplying 12 by 5, you are essentially adding 12 to itself 5 times. This is what we performed in the first part of the original exercise:- Calculate each multiplication separately to streamline the process.- For example, \( 12 \cdot 5 \) equals 60.\(\) This means there are five groups of 12, resulting in a total of 60. Similarly, when you multiply 3 by 6, \( 3 \cdot 6 = 18 \), indicating you have 6 groups of 3. This will help in simplifying expressions later when combining with other arithmetic operations.

Subtraction

Subtraction is another fundamental arithmetic operation. It is used to determine the difference between two numbers. In other words, it shows how much more one number is compared to another.After performing the necessary multiplication in the exercise, the next step was subtraction. - Substitution helps simplify the expression by replacing complex parts with their simpler equivalents. - For example, by substituting \( 12 \cdot 5 \) with 60 and \( 3 \cdot 6 \) with 18, you transform the complex expression into \( 60 - 18 \).From here, perform the subtraction process. Subtraction can be visualized as removing objects from a group. If you have a group of 60 and remove 18, you end up with 42, which is the solution to the expression.

Arithmetic Operations

Arithmetic operations are the rules that govern the world of numbers. They include addition, subtraction, multiplication, and division. When simplifying expressions, these operations must be performed in a specific order, known as the order of operations. The order is 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 multiplication took precedence over subtraction due to the order of operations. This is essential in simplifying expressions accurately: - Calculate the multiplication operations first. - Then perform the subtraction. Understanding and applying these operations ensures clarity and precision in solving mathematical problems. This simplifies what might seem complex, enabling you to focus on solving rather than getting tangled in the operations.