Problem 123 - Prealgebra and Introductory Algebra

Question

Evaluate the expression \(x \div y\) for the given values of \(x\) and \(y.\) $$x=-64.05, y=-6.1$$

Step-by-Step Solution

Answer

The result of the expression for the given values of \(x\) and \(y\) is 10.5

Step 1: Identify the values of x and y

The given values are \(x = -64.05\) and \(y = -6.1\) which are both negative numbers.

Step 2: Execute the Operation

Since the given operation is division, divide \(x\) by \(y\). That is, calculate the result of the expression \(-64.05 \div -6.1\).

Step 3: Calculation

The calculation gives \((-64.05) \div (-6.1) = 10.5\) This is the final result.

Key Concepts

Negative Numbers

When we talk about negative numbers, we refer to numbers that are less than zero. They often have a minus sign ("-") in front of them. Negative numbers can be found in everyday situations like measuring temperatures below zero or owing money. Understanding how negative numbers work is crucial, especially in math.

• In mathematics, two negative numbers when multiplied or divided always give a positive result.
• The sign rule for division is:
  • Positive divided by Positive = Positive
  • Positive divided by Negative = Negative
  • Negative divided by Negative = Positive
  • Negative divided by Positive = Negative

So, when we divide -64.05 by -6.1, since both are negative, the answer is positive. It's essential to keep these sign rules in mind when working with negative numbers and division.

Evaluate Expressions

Evaluating expressions involves calculating the value of a mathematical phrase. This means taking an expression like \(x \div y\) and finding its numerical result given specific values for variables (in this case, \(x = -64.05\) and \(y = -6.1\)). Evaluating helps simplify problems and understand relationships between numbers.To evaluate an expression:

• Substitute the given values into the expression.
• Follow the order of operations (PEMDAS/BODMAS) if necessary.
• Calculate to find the answer.

For our problem, once we substitute, it becomes a straightforward division: \(-64.05 \div -6.1\). No additional operations are needed in this case.

Prealgebra

Prealgebra lays the foundation for understanding basic arithmetic operations and their properties. It's often the stepping stone to more advanced math topics. A central part of prealgebra is understanding how to work with integers, fractions, and decimals effectively. Key topics in prealgebra include:

• Addition, subtraction, multiplication, and division of numbers.
• Understanding and applying the rules for integers.
• Manipulating decimals and negative numbers.

In this exercise, dividing two negative decimal numbers falls under prealgebra. By mastering such operations, you set a strong foundation for algebra and beyond.

Step by Step Solution

Going through mathematical problems step by step is vital in understanding how to solve them. A systematic approach can break down even complex problems into manageable parts.When tackling a step by step solution:

• Identify the given values and the operation needed.
• Perform the operation based on the rules (like sign rules for division of negative numbers)
• Calculate to find the result.

In our example, we:1. Identified \(x = -64.05\) and \(y = -6.1\).2. Recognized the operation as division.3. Performed the division, knowing two negatives make a positive, leading to \(10.5\).By following each step attentively, we can ensure accuracy and build better problem-solving skills.