Problem 100 - Prealgebra and Introductory Algebra

Question

Evaluate the expression for the given values of the variables. $$a \div(-b), \text { for } a=-36 \text { and } b=-4$$

Step-by-Step Solution

Answer

The final result of the evaluation of the given expression \(a \div(-b)\) for a = -36 and b = -4 is -9.

Step 1: Understand the expression

The given expression to evaluate is \(a \div(-b)\). According to the problem, a = -36 and b = -4. We need to substitute these values into the expression.

Step 2: Substitute the values

Substitute a = -36 and b = -4 into the expression. This gives us the new expression: -36 ÷ (-(-4)).

Step 3: Simplify the expression

Simplify the expression by removing the double negative in the denominator to get: -36 ÷ 4.

Step 4: Perform the division

Finally, perform the division. The result of -36 ÷ 4 is -9.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are crucial in mathematics, especially when dealing with losses, temperatures below zero, or depths below sea level. You'll often encounter negative numbers in real-world situations as well as in various mathematical problems.

When using negative numbers in arithmetic operations, it's essential to understand their rules. For example, when you multiply or divide two negative numbers, the result is positive. This knowledge is crucial in evaluating expressions and performing calculations. In the given exercise, both the values of \( a \) and \( b \) are negative. This affects the operation because dividing a negative number by another negative number will yield a positive result.

Expression Evaluation

Evaluating an expression involves replacing variables with given numbers and performing the necessary arithmetic operations to find the result. This process requires following the order of operations, which ensures that calculations are performed correctly.

In our exercise, the main expression to evaluate is \( a \div (-b) \). The first step involves substituting \( a \) and \( b \) with their respective values. By correctly plugging in these numbers, you can transform an algebraic expression into an arithmetic computation that can be solved to get the answer.

Substitution

Substitution is a fundamental concept in algebra where we replace variables with numbers. This method helps convert an expression with variables into something more tangible that you can compute.

• Identify the variables in the expression. In our exercise, these are \( a \) and \( b \).
• Replace the variables with the given numerical values.

In this particular problem, substitute \( a = -36 \) and \( b = -4 \). The expression \( a \div (-b) \) then becomes \(-36 \div (-(-4))\). By substitution, the expression changes from an abstract variable expression into a concrete arithmetic problem.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. Each of these operations has its own rules, especially when dealing with negative numbers.

• Addition/Subtraction: Adding a negative number is like subtracting, and subtracting a negative number is like adding.
• Multiplication/Division: When multiplying or dividing numbers, the result is positive if both numbers have the same sign and negative if they have different signs.

In this exercise, we focus on division. Specifically, we divide \(-36\) by \(4\) after simplifying from \(-(-4)\) to \(4\). Understanding how division works with negative numbers is key to arriving at the correct answer of \(-9\).