Problem 63 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ \frac{-3(4-2)+(-3)(-6)}{-4} $$

Step-by-Step Solution

Answer

Answer: -3

Step 1: Simplify the numerator

According to the order of operations, we have to carry out the operations inside the parentheses first, then perform multiplications and divisions. So the numerator is: $$ -3(4-2)+(-3)(-6) $$ Perform operations inside the parentheses first: $$ -3(2)+(-3)(-6) $$ Now, perform multiplications: $$ -6+18 $$ Finally, perform addition: $$ 12 $$ So, the simplified numerator is 12.

Step 2: Simplify the denominator

The denominator of the given expression is already simplified: $$ -4 $$

Step 3: Divide the simplified numerator by the simplified denominator

Now that we have simplified both the numerator and the denominator, we can divide them to find the value of the whole expression: $$ \frac{12}{-4} $$ Divide 12 by -4: $$ -3 $$ So, the value of the expression is -3.

Key Concepts

Numerical Expressions

Numerical expressions are a fundamental part of mathematics, especially in algebra and arithmetic. They consist of numbers combined with operation symbols such as addition, subtraction, multiplication, and division. When you're asked to evaluate a numerical expression, your main task is to simplify it to find its value.

• Each part of the expression needs special attention.
• Operations may need to be performed in a specific order, which is outlined by the order of operations.

Start from any enclosed expressions, such as those in parentheses, and progress towards the final operation. Simplifying expressions is like a step-by-step puzzle that requires orderliness and precision. It's essential to understand what each number and symbol means to navigate through the calculation successfully.

Arithmetic Operations

Arithmetic operations are the building blocks for any numerical expression. They include addition, subtraction, multiplication, and division. Each operation has its particular function and role in forming and solving equations and expressions.

• Addition combines numbers to increase total value.
• Subtraction takes away from one number using another.
• Multiplication "scales" numbers by a factor of another number.
• Division "splits" numbers into specified equal parts.

Understanding these operations is vital for accurately solving expressions. The correct order, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), ensures that you tackle expressions methodically. In your given expression, notice how multiplication is handled before addition even when the multiplication involves negative numbers.

Division of Integers

Division of integers involves splitting one whole number by another and it can sometimes result in negative or positive outcomes. This operation is crucial in reducing complex expressions into unique whole values through simple steps. Here’s what you should consider:

• If both integers are positive or both are negative, the result is positive.
• If one integer is positive and the other is negative, the result is negative.

In the solution provided, dividing the numerator 12 by the denominator -4 yields -3. This negative outcome is because the numerator is positive while the denominator is negative. Consistency in applying these rules is key to avoiding errors in calculations involving integers. It's all about systematically applying principles to ensure each integer division works correctly, regardless of the complexity of the expression.