Problem 125 - Elementary Algebra

Question

Problem: Evaluate: \(-6-(-2)(5)\) $$ \begin{aligned} \text { Incorrect Answer: } &-6-(-2)(5) \\ &=-6-10 \\ &=-16 \end{aligned} $$

Step-by-Step Solution

Answer

-6 - (-2)(5) = 4

Step 1: Understand the Problem

Evaluate the expression \( -6 - (-2)(5) \). Pay attention to the operations and signs.

Step 2: Apply the Rule for Subtraction

Recall that subtracting a negative number is the same as adding the positive. So, \(-(-2)\) becomes \(+2\). Rewrite the expression: \(-6 + 2(5)\).

Step 3: Multiply

Perform the multiplication inside the parentheses first. \(2 \times 5 = 10\). Rewrite the expression: \(-6 + 10\).

Step 4: Add

Add \(-6\) and \10\ together. \(-6 + 10 = 4\).

Key Concepts

Order of Operations

When evaluating algebraic expressions, it's important to follow the 'Order of Operations'. This ensures that expressions are evaluated consistently and correctly. The common acronym for this order is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In the context of our problem \( -6 - (-2)(5) \), it's crucial to follow these steps:

• First, handle parentheses: \( -(-2) \) becomes \( +2 \).
• Second, perform multiplication or division as they appear from left to right: \( +2 \times 5 \).
• Lastly, carry out addition or subtraction from left to right: \( -6 + 10 \).

By following PEMDAS, we maintain accuracy in our calculations and arrive at the correct answer.

Negative Numbers

Negative numbers can be tricky, but understanding how to work with them is crucial in algebra. A negative number is any number less than zero, and it's represented with a minus sign (e.g., \(-3\)). When dealing with negative numbers in operations:

• Subtracting a negative number is the same as adding its positive counterpart. For example, \( -(-2) \) becomes \( +2 \), as seen in our step-by-step solution.
• Multiplying two negative numbers results in a positive number: \( -2 \times -2 = 4 \).
• Conversely, multiplying a negative number by a positive results in a negative number: \( -2 \times 5 = -10 \).

In our problem, recognizing that \( -(-2) \) turns into \( +2 \) is crucial to follow through the correct order of operations and arrive at the precise solution.

Arithmetic Operations

Arithmetic operations are the basics of math that include addition, subtraction, multiplication, and division. Let's break them down:

• Addition: Combining two numbers to get a sum. For example, \( 3 + 5 = 8 \).
• Subtraction: Taking one number away from another. For instance, \( 7 - 4 = 3 \).
• Multiplication: Repeated addition of a number. For example, \( 4 \times 3 \) is the same as \( 4+4+4 = 12 \).
• Division: Splitting a number into equal parts. For instance, \( 12 \div 3 = 4 \).

In our problem, the key steps involved:

1. Turning \( -(-2) \) into \( +2 \).
2. Multiplying \( +2 \) by \( 5 \) to get \( +10 \).
3. Finally, adding \( -6 \) and \( +10 \) which equals \( +4 \).

By understanding and correctly applying these arithmetic operations, we effectively evaluated the original expression step-by-step.