Problem 48 - Elementary Algebra

Question

Simplify. $$ (-1-3) 2-15-3 \cdot(-7+22)-5 $$

Step-by-Step Solution

Answer

The simplified expression is -73.

Step 1: Simplify inside the parentheses

First, simplify the expressions inside each set of parentheses. For (-1 - 3), we calculate -1 - 3 = -4. For (-7 + 22), we calculate -7 + 22 = 15. This simplifies our equation to:\[(-4) imes 2 - 15 - 3 imes 15 - 5\]

Step 2: Perform multiplication operations

Next, perform the multiplication operations:Compute (-4) imes 2 = -8 and 3 imes 15 = 45. Substitute these values back into the expression:\[-8 - 15 - 45 - 5\]

Step 3: Perform subtraction operations

Finally, perform the subtraction operations. Start with the leftmost subtraction and proceed to the right: 1. Calculate -8 - 15 = -23. 2. Then, -23 - 45 = -68. 3. Lastly, -68 - 5 = -73. Therefore, the simplified expression is -73.

Key Concepts

Arithmetic Operations

In mathematics, arithmetic operations are basic calculations we use in everyday life. They include addition, subtraction, multiplication, and division. These operations allow us to manipulate numbers in a variety of helpful ways.

• Addition: Combining two or more numbers to get a total. For example, adding 2 and 3 gives you 5.
• Subtraction: Taking one number from another to find the difference. For example, subtracting 5 from 10 gives you 5.
• Multiplication: Repeated addition of the same number. For example, multiplying 4 by 3 means adding 4 three times, resulting in 12.
• Division: Splitting a number into equal parts. For example, dividing 10 by 2 means splitting 10 into two parts of 5 each.

Understanding these operations is crucial for simplifying expressions, as they form the foundation of more complex calculations.

Order of Operations

The order of operations is a set of rules that tells us the correct sequence to follow when evaluating a mathematical expression. This is important because different orders can lead to different answers. We often use the acronym PEMDAS to remember the order:

• Parentheses: Solve expressions inside parentheses first.
• Exponents: Calculate powers or roots next.
• Multiplication and Division: Perform these operations from left to right.
• Addition and Subtraction: Finally, handle these from left to right.

In our original problem, we follow the order of operations by first simplifying the expressions inside parentheses, then performing the multiplication, and lastly executing the subtraction. This ensures accuracy and consistency in solving the expression.

Algebraic Expressions

Algebraic expressions include numbers, variables, and arithmetic operations together. They can model real-world problems by using symbols to represent unknown values or quantities. Let's break it down into the key components of algebraic expressions:

• Numbers: Known quantities. For example, in the expression 3x + 2, the numbers are 3 and 2.
• Variables: Letters like x or y that represent unknown values or quantities.
• Operations: Combine numbers and variables with arithmetic operations to form expressions. For instance, 3x + 2 combines multiplication and addition.

In our task, we're focusing on simplifying a numerical expression, but similar steps apply when dealing with algebraic expressions. Understanding how numbers and operations work together helps us manipulate and simplify these expressions effectively.