Problem 101 - 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 value of the expression for \(a=-36\) and \(b=-4\) is 9.

Step 1: Substitute the given values into the expression

Firstly, replace \(a\) and \(b\) with their respective values in the arithmetic expression, \((-a) \div(-b)\). This gives us \((--36) \div (--4)\), because \(a=-36\) and \(b=-4\).

Step 2: Simplify the double negative

Two negatives make a positive in arithmetic. Therefore, the expression \((--36) \div (--4)\) becomes \(36 \div 4\).

Step 3: Perform the division

The value of \(36 \div 4\) is 9.

Key Concepts

Substitution in Expressions

Substitution in expressions is a fundamental step in evaluating algebraic expressions. This process involves replacing the variables in an expression with specific values that are given. In the case of our original exercise, this means simply taking the given values for variables and inserting them into the expression.

• Identify the variables in the expression. Here, you have variables \( a \) and \( b \).
• Check the values provided for each variable. For this exercise, \( a = -36 \) and \( b = -4 \).
• Substitute these values into the expression \((-a) \div (-b)\).
• Thus, the expression transforms to \((--36) \div (--4)\).

Remember that each time you substitute a variable with its value, you maintain the structure of the expression but replace the letters with numbers. Getting familiar with substitution is helpful as it is a key skill in algebra for evaluating expressions.

Arithmetic Simplification

Arithmetic simplification involves making an expression as simple as possible while maintaining its value. After substitution, we often encounter parts of expressions that can be reduced or simplified further.

Simplifying Double Negatives

Double negatives can appear tricky, but in arithmetic, they are straightforward. Two negative signs encountered consecutively translate to a positive. In our example, after substituting \(-a\) for \(-(-36)\), it becomes simply \(36\), and similarly, \(-b\) turns to \(-(-4)\) which simplifies to \(4\).

• Recognize that two negatives in succession will negate each other.
• Thus, \((--36)\) changes to \(36\), and \((--4)\) changes to \(4\).

The aim is to reduce unnecessarily complex expressions into simpler, equivalent ones that are more manageable. Once simplified, calculations become easier and less error-prone. This practice not only makes mathematics more approachable but also paves the way for solving more advanced problems.

Division in Algebra

Division is a critical mathematical operation, and understanding its functioning within algebra is essential. Once you have accurately substituted and simplified your expression, performing the division is usually the final step.

Performing the Division

To divide, you can think of it as sharing or distributing equally. In the algebraic context, once you have simple terms after subtraction and simplification, performing the division gives you the result of the expression.

• Check the simplified terms; here, it's \(36 \div 4\).
• Perform the arithmetic operation: \(36 \div 4 = 9\).
• This leads to the conclusion of the exercise with the final evaluated value.

Remember, division can sometimes lead to smaller values or fractions if the numbers are not perfectly divisible. Here, however, the result is an integer as \(36\) divides evenly by \(4\), giving a clean result of \(9\). Practicing division within different contexts reinforces your ability to handle various types of arithmetic expressions in algebra.