Problem 107 - Introductory Algebra for College Students

Question

A subtraction is expressed as addition of an opposite. Find this sum, indicated by a question mark. -8-(-13)=-8+13=?

Step-by-Step Solution

Answer

The sum of -8 and -(-13) is 5.

Step 1: Identify the operation

The first step is to identify the operation to be performed. In this case, it's -8-(-13). The operation to perform here is subtraction, but as the problem doesn't ask to subtract a positive number but a negative one, it requires to change the operation to addition.

Step 2: Convert the operation

The next step is to convert the operation from subtraction to addition. It should be remembered that subtracting a negative number is the same as adding a positive number. So, -8-(-13) can be taken as -8+13.

Step 3: Perform the operation

Finally, now the operation is a simple addition of two numbers, one negative and another positive. According to the rules of addition for negative and positive numbers, find the difference between the absolute values of the two numbers and give the resulting number the sign of the larger one. So, -8 + 13 results in 5.

Step 4: Review the final result

After performing the operation, the final result is 5. This is the sum we were looking for.

Key Concepts

Addition of Integers

Addition in algebra forms the basis for understanding how numbers interact. Whether working with positive or negative integers, the process is all about combining amounts. When you add integers, consider these points:

• The number line can help visualize integer addition. Positive numbers move you right, while negative numbers move you left.
• Addition is commutative, meaning changing the order of numbers does not matter. For example, \(3+(-5) = (-5)+3\).
• Positive integers increase a value, and negative integers decrease it.

To solve the exercise we first need to understand that -8 and +13 fall on this number line, which helps us conclude in the next steps. By using these basics, we can easily understand how integers come together to form a given result.

Subtraction as Addition

In introductory algebra, subtraction is not just taking away; it can be transformed into addition. The concept of "adding the opposite" is crucial. Here are some key things to remember:

• Subtraction of a number is the same as adding its negative counterpart. For instance, \(-7 - 4\) becomes \(-7 + (-4)\).
• If you subtract a negative number, you are effectively adding a positive number. So, \(-8 - (-13)\) becomes \(-8 + 13\).
• This transformation is due to the nature of inverse operations: subtraction is the inverse of addition.

Understanding this idea makes math problems with subtraction simpler, as it often boils down to an addition problem.

Negative Numbers

Negative numbers represent a value below zero on the number line. They can be a bit tricky at first, but there are some tools and rules that make working with them easier:

• Negative numbers are the opposite of positive numbers. On a number line, they are located to the left of zero.
• When you add a larger negative to a smaller positive, the result is still negative. When you add a larger positive to a smaller negative, the result is positive, as with our exercise \(-8 + 13 = 5\).
• Subtracting negatives can often make a result more positive, which is counterintuitive but essential to grasp.

By grasping these concepts, manipulating equations involving negative numbers in algebra becomes a lot more manageable. They are powerful yet require understanding fundamental rules.