Problem 33 - Elementary Algebra

Question

For the following exercises, perform the indicated operations. $$ -2-(-10) $$

Step-by-Step Solution

Answer

Question: Evaluate the expression: -2 - (-10) Answer: 8

Step 1: Remember the rule for subtracting negative numbers

When you subtract a negative number, it is the same as adding the positive version of that number. So, in our case, subtracting -10 is the same as adding 10:$$ -2 - (-10) = -2 + 10 $$

Step 2: Add the numbers

Now that we have changed the operation to addition, we can simply add the two numbers:$$ -2 + 10 = 8 $$ So, the result of the operation is 8.

Key Concepts

Subtraction of Negative Numbers

Subtracting negative numbers can be a bit confusing at first. However, it's a lot simpler when you think about it in terms of turning subtraction into addition. When you see problems like \(-2 - (-10)\), there's an essential rule to remember:

• Subtracting a negative is the same as adding a positive.

In practice, if you're asked to subtract \(-10\) from \(-2\), you can instead think of it as adding 10 to \(-2\).
This is because the minus sign in front of the negative number essentially cancels it out, making it positive. So, we rewrite our original question as \(-2 + 10\), which changes the operation from subtraction to addition, making it easier to solve.

Addition of Integers

Adding integers involves combining positive and negative numbers. Unlike subtraction, which can sometimes seem more complex, addition follows the straightforward rule of combining numbers:

• If both numbers are positive, the result is a straightforward addition.
• If both numbers are negative, you combine them as if they were positives, but the result is negative.
• If one number is negative and the other is positive, view it as a subtraction, where you take away the smaller number from the larger one.

In our original problem, after changing the operation to addition with \(-2 + 10\), we simply combine these integers.
Since 10 is positive and larger in absolute terms than 2, the result is positive, leading us to 8.
This type of integer operation is crucial because it allows for understanding more complex mathematical expressions and equations.

Basic Algebra Concepts

Basic algebra involves some fundamental principles that help deal with mathematical operations efficiently. Understanding these principles lays the foundation for further learning in mathematics.

• Order of Operations: Always perform calculations in the correct order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), often remembered by the acronym PEMDAS.
• Variables and Constants: Variables are symbols that represent numbers. Constants are fixed values. In expressions, understand the role each plays.
• Simplifying Expressions: This involves combining like terms and using distributive properties to make expressions easier to work with.

For our example problem, understanding how to manipulate and simplify operations like \(-2 - (-10)\) using the rules of subtraction and addition exemplifies basic algebra. Recognizing that subtraction can be converted into addition is an essential skill, often used to make solving equations more straightforward.