Problem 87 - Intermediate Algebra

Question

In the following exercises, simplify each expression. (a) \(27-(-18)\) (b) \(27+18\)

Step-by-Step Solution

Answer

Both expressions simplify to 45.

Step 1: Understand the Problem

The problem requires simplifying two different expressions: (a) involves subtraction with a negative number, and (b) involves addition.

Step 2: Simplify Expression (a)

The first expression is given as: i) \(27 - (-18)\)(ii)To simplify, recall that subtracting a negative number is the same as adding the absolute value of that number: (iii)\(27 - (-18) = 27 + 18\)(iv)(iiiiv)(AB(iv)) \(27 + 18 = 45\)

Step 3: Simplify Expression (b)

The second expression is: (i)\(27 + 18\)(ii)Simply add the two numbers together: (iii)\(27 + 18 = 45\)

Step 4: Verify the Results

Both parts (a) and (b) result in the same value. Therefore, the simplified expressions for both are 45.

Key Concepts

Negative Numbers

Negative numbers can often be confusing, but understanding them is crucial in algebra. A negative number is any number less than zero and is represented with a minus sign (-) in front of it. For example, -5 is five units less than zero. Negative numbers are used to represent losses, decreases, or values below a defined zero-point. When dealing with negative numbers in algebra, it’s important to remember that subtracting a negative is equivalent to addition.

For instance, in our exercise, the expression:

- Given: \(27 - (-18)\),
- Converting subtraction of a negative to addition: \27 + 18\,
- Finally: \(27 + 18 = 45\)

This conversion is essential in simplifying expressions accurately.

Subtraction

In algebra, subtraction means taking away one quantity from another. Subtraction with negative numbers can be tricky, but it’s essential to get it right. Let’s break it down:

When you subtract a negative number, it’s the same as adding the positive equivalent of that number. This is because two negatives cancel each other out. Consider our exercise:

- Expression: \(27 - (-18)\),
- This converts to: \27 + 18\

Here, subtracting -18 is the same as adding 18. This principle helps in simplifying complex expressions.

Addition

Addition in algebra is straightforward but can become complex when combined with negative numbers and subtraction. Let's go through some key points using our second expression:

- Expression: \(27 + 18\)

Addition simply combines the two quantities. In our example:

- Given: \27 + 18\,
- Calculation: \27\ plus \18\ equals \45\

Adding positive numbers is simply counting up, but being comfortable with transitioning from subtraction involving negatives is key to algebraic manipulation. This skill lays the foundation for solving more complex problems.