Problem 53 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ 3(4-6) $$

Step-by-Step Solution

Answer

Question: Evaluate the expression: 3(4 - 6) Answer: -6

Step 1: Identify the subtraction operation inside the parentheses

The mathematical expression consists of a subtraction operation 4-6 inside the parentheses.

Step 2: Perform the subtraction operation

To find the value inside the parentheses, subtract the numbers: 4 - 6 = -2.

Step 3: Solve the multiplication

Now, multiply the result from step 2 (-2) by the number outside the parentheses (3): 3 × (-2) = -6. Therefore, the given expression evaluates to -6.

Key Concepts

Order of Operations

When evaluating expressions, it's essential to follow the correct order of operations. This is a set of rules that mathematicians have agreed upon to avoid confusion when performing calculations that involve more than one operation, such as addition, subtraction, multiplication, or division. The order of operations is often remembered by the acronym PEMDAS:

• Parentheses: Perform calculations inside parentheses first.
• Exponents: Solve expressions with exponents after any parentheses are taken care of.
• Multiplication and Division: Carry out multiplication and division from left to right, whichever comes first.
• Addition and Subtraction: Finally, handle addition and subtraction from left to right, whichever comes first.

In the given exercise, we see this rule in action. We first address the operation inside the parentheses (4-6), and only afterward, do we multiply by 3. Ignoring this hierarchy could lead to incorrect results and confusion in more complex expressions.

Arithmetic Operations

The base of mathematics is formed by arithmetic operations, which include addition, subtraction, multiplication, and division. In the context of the exercise provided, we encounter two of these operations:

• Subtraction: Subtracting one number from another is the process of finding the difference between them. In our example, 4-6, we find that the difference is negative because we are taking a larger number (6) away from a smaller one (4), resulting in -2.
• Multiplication: Multiplication is essentially repeated addition. When we multiply 3 by -2, we are adding -2 together three times, leading to -6.

Grasping these basic operations is crucial as they are the building blocks for solving more advanced problems in algebra and beyond.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations. In this exercise, although there are no variables, the structure of 3(4-6) is an example of an algebraic expression where arithmetic operations need to be applied to evaluate its value.

When we evaluate algebraic expressions, we are essentially replacing the variables with numbers (if there are any) and calculating the result following the order of operations. By mastering the evaluation of such expressions, students can gradually understand more complex situations like solving equations, working with functions, and analyzing graphs. The goal is to become comfortable with combining arithmetic to find the value of these expressions, just as we did step by step with the provided problem.