Problem 42 - Elementary Algebra

Question

Find the sums. \(-9+(-6)\)

Step-by-Step Solution

Answer

Answer: -15.

Step 1: Rewrite as Addition

Since both numbers are negative, we can rewrite the expression as an addition problem: \(-9 + (-6) = -9 - 6\)

Step 2: Add the Magnitudes

Next, we add the magnitudes (ignoring the negative signs) of the two numbers: \(9 + 6 = 15\)

Step 3: Determine the Sign

Since both numbers were negative, the sum will also be negative.

Step 4: Write the Final Answer

Combining the sum with the negative sign, the final answer is: \(-15\)

Key Concepts

Algebraic Addition

When dealing with algebraic addition, it's essential to understand how to combine numbers that include negative values. Impressively simple yet puzzling at first glance, algebraic addition requests our attention to both the magnitude and the sign of the numbers involved.

For instance, when adding negative numbers, you start by removing the signs to deal with the pure numbers. Addition often has the underpinning principle of combining quantities, but when both operands are negative, it's as if you're gathering a debt, more and more, leading to a larger amount owed. This perspective is why the sum of two negative numbers is a bigger negative number.

In the textbook problem \( -9 + (-6) \), the process is to first look at the magnitudes: 9 and 6. Then add them together to get 15. Since both original numbers were negative, the combined sum is also negative, resulting in \( -15 \). This exercise becomes a quick mental calculation with practice, folding both magnitude and direction (sign) into a single swift operation.

Additive Inverses

Delving into the concept of additive inverses, or as some might call it, the 'opposites', unveils a fascinating aspect of number theory within the bigger algebraic picture. An additive inverse is essentially a number which, when added to the original number, yields zero—the very definition of cancellation.

The additive inverse of a negative number is its positive counterpart and vice versa. For the number \( -9 \), the additive inverse is \( 9 \), since \( -9 + 9 = 0 \). Similarly, for \( -6 \), the additive inverse is \( 6 \). This concept is central to solving equations and also understanding that each number holds the potential to nullify itself with its exact inverse.

Understanding additive inverses gives learners a stronger grasp of balancing equations and simplifying expressions, making it easier to deal with subtraction by reimagining it as adding the inverse.

Integer Operations

Integer operations cover the gamut of mathematical processes including addition, subtraction, multiplication, and division with whole numbers which can be positive, negative, or zero. These operations follow specific rules that help maintain order and relation between the numbers in the world of integers.

For addition, as in the case of the exercise \( -9 + (-6) \), the operation combines both values taking into consideration their signs to reach a sum that accurately represents the total. We can think of this like temperature change; adding a decrease of 9 degrees to a decrease of 6 degrees results in a total decrease of 15 degrees, or, in this case, \( -15 \).

These operations aren't just abstract concepts; they model real-world situations like financial transactions involving debt and credit, temperature variations, and altitude changes. Mastering integer operations allows students to nimbly navigate both practical and theoretical scenarios where whole numbers play a vital role.