Problem 80 - Intermediate Algebra

Question

Simplify each numerical expression. $$ -65 \div 5-(-13)(-2)+(-36) \div 12 $$

Step-by-Step Solution

Answer

The expression simplifies to \(-42\).

Step 1: Division - First Term

Simplify the first division in the expression: \(-65 \div 5\). Calculate: \[-65 \div 5 = -13\]. Now the expression becomes: \[-13 - (-13)(-2) + (-36) \div 12\].

Step 2: Division - Third Term

Simplify the division in the expression: \(-36 \div 12\). Calculate: \[-36 \div 12 = -3\]. Now the expression becomes: \[-13 - (-13)(-2) - 3\].

Step 3: Multiplication

Simplify the multiplication in the expression: \((-13)(-2)\). Calculate: \[(-13)(-2) = 26\]. Now the expression becomes: \[-13 - 26 - 3\].

Step 4: Subtraction

Carry out the subtraction from left to right: \(-13 - 26\). Calculate: \[-13 + (-26) = -39\]. Now the expression becomes: \[-39 - 3\].

Step 5: Final Subtraction

Complete the final subtraction: \(-39 - 3\). Calculate: \[-39 + (-3) = -42\]. This is the simplified result of the expression.

Key Concepts

Division in Algebra

Division in Algebra involves dividing one number, the dividend, by another, the divisor. In algebraic expressions, this process remains consistent with basic arithmetic rules. When dividing integers, the sign of the result depends on the signs of the numbers involved. For example, when you divide

• two numbers with the same signs, the result is positive,
• whereas with different signs, the result is negative.

In our example, \[ -65 \div 5 \] explains the division process: Here, - -65 (dividend) is negative, - and 5 (divisor) is positive.
The result is negative, resulting in \( -13 \). Leading to a simplified expression: \[ -13 - (-13)(-2) + (-36) \div 12 \].
Division in algebra assists in reshaping expressions, especially to facilitate subsequent operations.

Multiplication of Integers

In mathematics, Multiplication of Integers extends beyond simple arithmetic principles. Importantly, the rule concerning signs remains identical:

• The product of two integers with the same sign is positive,
• while the product of integers with different signs is negative.

Consider \[ (-13) \times (-2) \] in our exercise.
Both integers are negative, thus their product equals a positive number \( 26 \).
This step transforms the expression into \( -13 - 26 - 3 \), making subsequent operations straightforward.

Order of Operations

Understanding the Order of Operations is key to solving expressions correctly. In mathematics, this orderly method ensures calculations follow predictable outcomes. The standard order is often remembered using the acronym PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and roots, etc.)
• M: Multiplication and D: Division (left to right)
• A: Addition and S: Subtraction (left to right)

In the given exercise, after simplifying divisions (step 1 and step 2), we proceed with Multiplication (step 3), then handle Subtraction (in steps 4 and 5).
Adhering to this order prevents calculation errors, guiding clear and logical problem-solving steps.

Integer Arithmetic

Integer Arithmetic encompasses operations involving whole numbers, inclusive of negative, positive integers, and zero.
Basic operations include:

• Addition
• Subtraction
• Multiplication
• Division

Each operation adheres to specific numerical rules. Understanding these rules creates clarity in algebraic processes.
In our solution example: 1. We divided integers with different signs resulting in a negative integer first. 2. We multiplied negative integers, yielding a positive result. 3. Finally, subtraction was employed as per integer arithmetic rules.
Thus, comprehending these fundamental rules simplifies evaluating complex expressions accurately.