Problem 39 - Elementary Algebra

Question

Translate each sentence to a mathematical statement and then simplify. Subtract 2 from the difference of 8 and \(5 .\)

Step-by-Step Solution

Answer

The simplified expression is 1.

Step 1: Identify the operation

The sentence asks to "subtract 2 from the difference of 8 and 5." The key operations are subtraction and finding the difference.

Step 2: Translate the difference part

The phrase "the difference of 8 and 5" means that we need to subtract 5 from 8. This can be written mathematically as: \[ 8 - 5 \]

Step 3: Incorporate the subtraction of 2

Now, we need to subtract 2 from the result obtained in Step 2. The mathematical expression becomes: \[ (8 - 5) - 2 \]

Step 4: Simplify the expression

First, simplify the inner expression by performing the subtraction:\[ 8 - 5 = 3 \]Then, subtract 2 from this result:\[ 3 - 2 = 1 \]

Key Concepts

Subtraction

Subtraction is a fundamental operation in mathematics that involves taking one quantity away from another. It helps us to find out how much is left or the balance between two quantities.
It is often represented by the minus sign (-). In our step-by-step example, subtraction plays a key role.
We are required to perform two subtraction operations:

• First, find the difference between 8 and 5.
• Then subtract 2 from the result.

Remember, in subtraction, the sequence of operations is important. Ensure that you always subtract the correct quantities.

Difference

When we refer to the term 'difference' in mathematics, we specifically mean the result of subtracting one number from another.
In the expression given, "the difference of 8 and 5" refers to the operation where 5 is subtracted from 8.
This results in the expression:

• \( 8 - 5 \)

Calculating differences helps us compare two numbers and gauge how much larger one is than the other.
It is an essential aspect of understanding numerical relationships in various contexts.

Mathematical Expression

A mathematical expression is a combination of numbers, operators, and sometimes variables that together denote a value.
They are crucial in translating words into equations that can be mathematically resolved, as seen in our step-by-step solution.
The phrase "subtract 2 from the difference of 8 and 5" translates into the mathematical expression:

• \((8 - 5) - 2\)

Mathematical expressions allow us to perform operations in a structured manner, ensuring clarity and accuracy.

Simplification

Simplification involves reducing a mathematical expression to its simplest form, making it easier to work with.
In our example, simplifying begins with solving the expression inside the parentheses:

• Calculate \(8 - 5\) to get 3.
• Then subtract 2 from this result, giving us \(3 - 2\).

The final simplified form is the number 1.
Simplification is a significant step in interpreting and solving mathematical problems, promoting efficiency and understanding.