Problem 107 - Intermediate Algebra

Question

In the following exercises, simplify each expression. $$ 26-3(2-7) $$

Step-by-Step Solution

Answer

11

Step 1: Simplify inside the parentheses

First, simplify the expression inside the parentheses: \(2 - 7 = -5\)

Step 2: Multiply by the number outside the parentheses

Next, multiply the simplified inside result by the outside number: \(-3 \times -5 = 15\)

Step 3: Subtract the result from the original number

Finally, subtract the product from 26: \(26 - 15 = 11\)

Key Concepts

Order of Operations

In mathematics, the order of operations is crucial for solving expressions correctly. It tells us the sequence in which to perform each operation so we can find the correct result. The typical order can be remembered using the acronym PEMDAS:

• P for Parentheses
• E for Exponents
• MD for Multiplication and Division (from left to right)
• AS for Addition and Subtraction (from left to right)

When using PEMDAS, always start with the highest priority operation. This means solving anything inside parentheses first, then exponents, and so forth. This order helps to ensure that everyone solves expressions in the same way and gets the same answer. For example, in the expression '26 - 3(2 - 7)', we follow these steps: first the parentheses, next the multiplication, and lastly the subtraction to get the correct answer.

Parentheses in Mathematics

Parentheses in mathematics signal which part of an expression should be evaluated first. Anything inside the parentheses must be simplified before any other operations are performed outside of them. This is why parentheses appear at the top of the PEMDAS hierarchy. Parentheses are particularly useful in complex expressions to clearly indicate which operations should be grouped together. For example, in the expression '26 - 3(2 - 7)', the part '2 - 7' is inside the parentheses. We simplify this to get '-5' before continuing with multiplication outside the parentheses. Without parentheses, an expression could be interpreted in multiple, incorrect ways.

Basic Arithmetic Operations

Basic arithmetic operations include addition, subtraction, multiplication, and division. Understanding these operations is foundational for solving more complex mathematics. Let's see how each one works within our example expression: '26 - 3(2 - 7)':

• Subtraction: First, we solve inside the parentheses: '2 - 7', which equals '-5'. This is a basic subtraction operation.
• Multiplication: Next, we multiply the result from the parentheses by the number outside, which is '-3'. So, '-3 * -5' equals '15'.
• Subtraction: Finally, we subtract this result from '26': '26 - 15', which leaves us with '11'. This is again a basic subtraction operation.

Each of these steps involves a basic arithmetic operation, demonstrating how fundamental these concepts are in solving expressions.