Problem 79 - Intermediate Algebra

Question

In the following exercises, simplify each expression. (a) \(13-7\) (b) \(-13-(-7)\) (c) \(-13-7\) (d) \(13-(-7)\)

Step-by-Step Solution

Answer

(a) 6 (b) -6 (c) -20 (d) 20

Step 1: Simplify 13 - 7

Start by subtracting 7 from 13. So, we calculate 13-7=6.

Step 2: Simplify -13 - (-7)

Simplify the expression by converting the double negative into a positive. This can be changed to -13 + 7. Simplify -13 + 7 which equals -6.

Step 3: Simplify -13 - 7

Subtract 7 from -13 to get -20. So, -13 - 7 = -20.

Step 4: Simplify 13 - (-7)

Simplify the expression by converting the double negative into a positive. This can be changed to 13 + 7. Simplify 13 + 7 which equals 20.

Key Concepts

arithmetic operations

When simplifying expressions, understanding arithmetic operations is essential. Arithmetic operations comprise four basic mathematical actions: addition, subtraction, multiplication, and division.
In this exercise, we mainly focus on addition and subtraction, which are often used in day-to-day math problems. Every arithmetic operation has properties that are crucial for solving equations correctly.
For instance, the commutative property of addition tells us that changing the order of numbers doesn't change the sum: \( a + b = b + a \). However, subtraction is not commutative: \( a - b eq b - a \).
These properties offer a foundation that facilitates better understanding and efficient problem-solving. By following them, we can simplify complex expressions step-by-step and avoid making mistakes.

negative numbers

Negative numbers can initially be confusing, but they are just as important as positive numbers. A negative number is a number less than zero, represented with a minus sign (\( - \)).
One key aspect of working with negative numbers is understanding how to add and subtract them properly.
When you have two negative numbers, such as \(-13 - (-7)\), converting a double negative into a positive can simplify the problem. This changes it into \(-13 + 7\). Essentially, subtracting a negative number is the same as adding its positive counterpart. It turns two minuses into a plus.
Additionally, subtracting a positive number from a negative number like \(-13 - 7\) means you are moving further left on the number line, resulting in a more negative number.

addition and subtraction

Adding and subtracting numbers correctly is fundamental in simplifying expressions. When performing addition and subtraction:

• Always make sure to perform operations within parentheses first, then follow the standard order of operations (PEMDAS/BODMAS).
• With two positive numbers like \(13 - 7\), you simply subtract the smaller number from the larger one to get the difference.
• For expressions like \(13 - (-7)\), change the subtraction of a negative number to addition, which makes it \(13 + 7 = 20\).
• When subtracting a larger positive number from a smaller positive number, you will get a negative result, such as \(-13 - 7 = -20\).

These examples show practical application of addition and subtraction rules. Understanding them deeply ensures you can tackle any similar problems with confidence and accuracy.