Problem 37 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ \frac{-39}{3} $$

Step-by-Step Solution

Answer

Answer: -13

Step 1: Identify the numerator and denominator

The given expression is: $$ \frac{-39}{3} $$ Here, the numerator is -39, and the denominator is 3.

Step 2: Perform the division

Now, we need to divide the numerator by the denominator: $$ \frac{-39}{3} = -39 \div 3 $$

Step 3: Calculate the result

Perform the division to find the value of the expression: $$ -39 \div 3 = -13 $$ Therefore, the value of the given expression is -13.

Key Concepts

Division of Integers

In algebra, the division of integers can seem perplexing, but it follows a straightforward rule: dividing two integers with the same sign results in a positive quotient, while dividing integers with different signs yields a negative quotient. Additionally, an integer divided by zero is undefined, as division by zero does not provide a meaningful answer in the realm of integers.

For instance, when dealing with the expression \( \frac{-39}{3} \) from the exercise, it's evident that we are dividing a negative integer (-39) by a positive one (3). According to the rules, since the signs are different, the result will be a negative integer. Understanding how signs affect the division outcome is crucial for simplifying algebraic expressions involving integers.

Numerators and Denominators

The concepts of numerators and denominators are at the heart of fractions and division in algebra. A fraction consists of a numerator, which is the number above the division line, and a denominator, which is the number below the line. The numerator represents how many parts you have, while the denominator indicates how many parts make up a whole.

The exercise \( \frac{-39}{3} \) serves as an excellent example of this fundamental principle. Here, -39 is the numerator, revealing that we are dealing with negative thirty-nine parts. Meanwhile, the denominator is 3, signifying these parts are divided into three equal groups. When simplifying such expressions, it's imperative to identify and understand the roles of the numerator and the denominator to accurately perform the division.

Arithmetic Operations in Algebra

Arithmetic operations in algebra, which include addition, subtraction, multiplication, and division, should be approached with the same strategy as when dealing with ordinary numbers. However, particular attention must be paid to the handling of variables and constants to preserve mathematical relationships.

Division, one of these operations, requires two numbers: the dividend (numerator) and the divisor (denominator). The simplification of \( \frac{-39}{3} \) illustrates a direct application of division, where we determine the quotient by dividing the dividend by the divisor. The solution steps are to first identify the values to be divided (as in Step 1) and then carry out the division (as in Step 2). The simplicity of operations in algebra stems from their consistency with familiar arithmetic, and mastering these will enable students to handle more complex algebraic expressions with confidence.