Problem 18 - Algebra for College Students

Question

Simplify each numerical expression. Be sure to take advantage of the properties whenever they can be used to make the computations easier. $$[63+(-87)]+(-64)$$

Step-by-Step Solution

Answer

The expression simplifies to -88.

Step 1: Simplify the Expression Inside the Brackets

The expression begins with a set of brackets: \([63 + (-87)]\). Within the brackets, we are adding the number 63 to -87. To simplify this, think of it as subtraction: \(63 - 87\). Performing this calculation gives us \(-24\).

Step 2: Simplify the Result with the Remaining Term

Now replace the bracketed expression with its result: \([-24] + (-64)\). Here, we need to add \(-64\) to \(-24\). This is equivalent to adding two negative numbers. Adding \(-64\) and \(-24\) results in \(-88\).

Key Concepts

Properties of Addition

The properties of addition play a crucial role in simplifying expressions, especially in arithmetic and algebra. They help us to rearrange, group, and combine numbers in ways that make calculations easier and more intuitive. Here are some key properties:

• Commutative Property: This property states that the order in which you add numbers does not change the sum. For example, \(a + b = b + a\). It means that when adding numbers, we can switch the order if it is more convenient.
• Associative Property: This property tells us that when three or more numbers are added together, the grouping of numbers does not affect the sum. For example, \((a + b) + c = a + (b + c)\). It can be useful when dealing with algebraic expressions as it allows flexible grouping of terms.
• Identity Property: The addition identity property states that any number plus zero is the number itself. For example, \(a + 0 = a\). This concept helps in simplifying expressions by eliminating zero terms.

Understanding these properties allows us to manipulate and simplify expressions more effectively, making them a fundamental part of solving mathematical problems.

Negative Numbers

Negative numbers can sometimes be confusing, but they are simply numbers less than zero. They often appear in real-world contexts, such as temperatures below freezing or drops in altitude. Here's how to effectively work with them:

• Adding Negative Numbers: When you add a negative number to a positive number, you are essentially subtracting the absolute value of the negative number. For example, adding \(-87\) to 63 is the same as 63 minus 87, yielding \(-24\).
• Combining Negative Values: When adding two negative numbers, such as \(-24\) and \(-64\), you're effectively adding their absolute values and then applying the negative sign. So, \(-24 + (-64)\) results in \(-88\) because 24 + 64 equals 88, with the negative sign indicating the direction on the number line.

By understanding these simple rules, working with negative numbers becomes less daunting and more intuitive. It is important to remember that negative signs indicate direction, and these signs play a critical role in simplifying expressions.

Algebraic Expressions

Algebraic expressions are a combination of constants, variables, and operations (such as addition, subtraction, multiplication, and division). They can be simplified or evaluated by following certain rules and using operations:

• Simplifying Expressions: To simplify an expression like \( [63 + (-87)] + (-64) \), the goal is to perform operations inside the brackets first, according to the order of operations. This often involves simplifying the terms step by step.
• Variables and Constants: Algebraic expressions often include variables, which are symbols representing numbers. Although they are not in our example, understanding how they work can help you deal with similar structures that involve variables.
• Like Terms: When simplifying expressions, it is important to combine like terms. These are terms that have the same variable part. Again, while our current example features only constants, recognizing this concept helps when you do encounter variables.

Mastering these basic techniques is crucial in working with algebraic expressions, enabling you to simplify complex problems and find solutions efficiently.