Problem 56 - Prealgebra

Question

Evaluate each expression if \(a=6, b=4,\) and \(c=5\). $$8 a-3 b$$

Step-by-Step Solution

Answer

The value of the expression is 36.

Step 1: Substitute Given Values

Start by substituting the values of the variables into the expression. The original expression is \(8a - 3b\). Given \(a = 6\) and \(b = 4\), substitute these values: \[8 \times 6 - 3 \times 4\].

Step 2: Perform Multiplications

First, calculate the multiplication operations: \(8 \times 6 = 48\) and \(3 \times 4 = 12\). This simplifies the expression to \(48 - 12\).

Step 3: Calculate the Result

Subtract the results obtained from the multiplications: \(48 - 12 = 36\).

Key Concepts

Evaluating Expressions

Evaluating expressions is the process of replacing variables within an algebraic expression with their numerical values and performing the necessary operations to simplify the expression to a single value. In our example, given the expression \(8a - 3b\) and values \(a = 6\) and \(b = 4\), the process involves direct substitution and arithmetic.

• First, substitute the given values into the expression: \(8 \times 6 - 3 \times 4\).
• Next, perform the operations in the order dictated by the expression. This often involves multiplication before subtraction, as per the order of operations (PEMDAS/BODMAS).
• Finally, calculate to find the numerical result: \(48 - 12 = 36\).

Breaking down expressions by evaluating them step by step is crucial. It helps in understanding how changes in variable values impact the result.

Prealgebra

Prealgebra is a foundational mathematical concept that prepares students for Algebra. It includes working with expressions, fundamental arithmetic operations, and understanding variables and their manipulation. In exercises like the one we've solved, identifying and substituting variables with given numerical values is central. Prealgebra encourages you to:

• Recognize patterns and relationships between numbers and operations.
• Understand that letters can represent numbers and solving expressions involves replacing these letters (variables) with numbers.
• Break down problems into manageable steps, which builds problem-solving skills.

Prealgebra serves as a bridge between basic arithmetic and the more abstract concepts found in algebra, making it a vital area of study for developing solid mathematical foundations.

Basic Arithmetic Operations

Basic arithmetic operations are the building blocks of mathematics. They comprise addition, subtraction, multiplication, and division. These operations are crucial when evaluating expressions.In the expression \(8a - 3b\):

• Multiplication is performed first: \(8 \times 6 = 48\) and \(3 \times 4 = 12\).
• Subtraction follows the multiplication: \(48 - 12\) results in the final answer of 36.

Understanding these operations and the order in which they are executed is key to successfully evaluating expressions. It's important to remember concepts like the order of operations to accurately perform calculations. When faced with challenges in arithmetic, practicing these basic operations can improve your skills and confidence in solving more complex problems.