Problem 18 - Algebra A Combined Function

Question

Subtract. \(-36-51\)

Step-by-Step Solution

Answer

The result of -36 - 51 is -87.

Step 1: Understand the Problem

The problem requires subtracting two integers: -36 and 51. The operation needed is subtraction of 51 from -36.

Step 2: Rewrite Subtraction as Addition

When we subtract a number, it is equivalent to adding its negative. Therefore, we can rewrite \(-36 - 51\) as \(-36 + (-51)\).

Step 3: Add the Two Negative Numbers

To add two negative numbers, add their absolute values and keep the negative sign. The absolute values are 36 and 51, which add up to 87.

Step 4: Apply the Negative Sign

Apply the negative sign to the sum. Since both numbers were negative, \(-36 + (-51) = -87\).

Key Concepts

Absolute Value

The absolute value of a number represents its distance from zero on the number line, without considering direction. It is always a positive quantity or zero. For example, the absolute value of both -36 and 36 is 36, written as \(|-36| = 36\) and \(|36| = 36\). The absolute value is vital when dealing with integer subtraction, especially when transforming a subtraction problem into an addition one. This transformation allows us to identify how far away one number is from another, disregarding their directions. This simplifies processing negative numbers, making it a foundational step in solving problems.

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). They are the opposite of positive numbers and are prevalent in calculations, indicating a deficit or loss. When performing operations with negative numbers, it’s crucial to understand that:

• Adding a negative number is equivalent to subtraction.
• Subtracting a negative number is like adding a positive number.

For instance, in the problem evaluated here, we deal with both -36 and -51. When these negative numbers are combined through addition, their combined magnitude becomes even larger in the opposite direction on the number line.

Addition of Integers

Integer addition is a straightforward arithmetic operation. However, when the integers have different signs, the operation becomes more nuanced. When adding two negative integers, like in our original problem (-36 + (-51)), we follow these steps:

• First, find the absolute values of the integers.
• Add these absolute values together.
• Finally, apply the negative sign to the sum, as both original integers were negative.

This process, resembling adding two positive integers and then affixing a negative sign, ensures accurate resolution of addition involving negative numbers.

Integer Operations

Integer operations include addition, subtraction, multiplication, and division. They are fundamental in mathematics and essential for various real-life applications. When performing integer subtraction, such as the conversion to addition of integers often simplifies the calculation. In the given exercise, we changed the subtraction of 51 from -36 into adding the negative of 51 to -36, thereby leveraging our understanding of integer addition. By mastering integer operations, especially transformation techniques like converting subtraction into addition, students can handle complex problems and real-world math challenges more effectively, increasing their mathematical confidence and capability.