Problem 16 - Algebra 2 and Trigonometry

Question

Use the definition of subtraction to write each subtraction as a sum. \(7-(-2)=9\)

Step-by-Step Solution

Answer

\(7 - (-2)\) as a sum is \(7 + 2 = 9\).

Step 1: Understanding Subtraction as Addition

Subtraction can be redefined as adding the opposite. This means that for any numbers \(a\) and \(b\), subtracting \(b\) is the same as adding the opposite of \(b\), which is \(-b\). Here we have \(7 - (-2)\).

Step 2: Rewrite the Expression

According to the rule of subtraction as addition, \(7 - (-2)\) can be rewritten as \(7 + 2\) because subtracting a negative number is the same as adding its positive.

Step 3: Perform the Addition

Now, we perform the addition: \(7 + 2 = 9\). This confirms our expression was rewritten correctly as the subtraction and addition yield the same result.

Key Concepts

Negative Numbers

Understanding negative numbers is crucial when dealing with integer operations, especially subtraction as addition. Negative numbers are numbers less than zero, represented by a minus sign (-) in front of the number, like

• -1
• -5
• -10

These numbers appear on the left side of the number line. For instance, if you think about temperatures, a reading of -5 degrees indicates a temperature below zero. Negative numbers can sometimes confuse students because they inherently want to deduct or take away, but when subtracted, the operation turns opposite.

Thus, subtracting negative numbers involves adding positive equivalents. It's vital to grasp this principle, as it simplifies complex calculations by converting them into more manageable ones.

Integer Operations

Integer operations include various mathematical processes performed on whole numbers. These operations primarily encompass addition, subtraction, multiplication, and division. Integer operations are fundamental to mastering basic arithmetic and later aspects of algebra. Here, we look at how subtraction can transform into addition when dealing with negative integers.

Think of integers as numbers without fractions, where the opposite nature of positive and negative values plays a crucial role. Subtracting an integer from another is often more straightforward when rewritten in its addition form. For example, taking away -2 from 7 (i.e., 7 - (-2) ), equates to adding 2 to 7 (i.e., 7 + 2 ). This underscores a handy rule: "subtracting a negative is like adding a positive."

Understanding these integer operations is key to excelling in mathematics and moving confidently through advanced problems.

Addition

Addition is one of the simplest and most frequently used operations in mathematics. It involves combining two or more numbers to find their total sum. The process of addition becomes even more interesting when we observe it in the context of negative numbers and subtraction. When turned into addition, subtraction problems, especially those involving negatives, become less intimidating and more logical.

For example, consider the equation 7 - (-2). By changing subtraction to addition, the equation becomes 7 + 2. This makes the operation much more straightforward as it turns a potentially confounding negative problem into an easy positive addition:

• Adding two positive numbers directly.
• Easily finding the sum by adding directly as normal.

In practical terms, learning this simplification can improve your problem-solving speed and accuracy. It's about understanding the relationship between addition and subtraction and using it to your advantage.