Problem 126 - Algebra and Trigonometry

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The difference between the product of six and a number and negative two times the number.

Step-by-Step Solution

Answer

The algebraic expression for the given English phrase is \(8x\).

Step 1: Identify the variable and translate the English phrase

In this case, the 'number' is \(x\). We can translate the phrase into the algebraic expression: 'the difference between the product of six and a number and negative two times the number' to \(6x - (-2x)\).

Step 2: Simplify the expression

We can simplify the expression further by resolving the double-negative situation. Therefore, \(6x - (-2x)\) simplifies to \(6x + 2x\).

Step 3: Combine like terms

Combine the like terms to simplify the expression. \(6x + 2x = 8x\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is all about breaking them down into their simplest form. It involves using mathematical operations to resolve elements like double negatives, which can be tricky at first. Let’s explore how we tackled this in our problem.

First, we translated the given phrase into an algebraic expression, but before simplifying it, we noticed a double negative. The phrase was 'the difference between the product of six and a number, and negative two times the number,' which converted to the expression \(6x - (-2x)\).

When simplifying an expression like this, the double negative can be resolved by converting \(-(-2x)\) into \(+2x\). Thus, \(6x - (-2x)\) became \(6x + 2x\). By addressing the double negative, we removed a potential source of confusion and moved closer to the expression's simplest form.

Simplifying expressions often involves steps like these, where we're converting troublesome elements into more manageable ones, paving the way for cleaner and more concise computations.

Combining Like Terms

The next critical step in solving algebraic expressions is combining like terms. 'Like terms' are terms within an expression that have the same variable raised to the same power, effectively letting us simplify the expression by consolidating them.

In our example, after simplifying \(6x + 2x\), we proceed to combine the like terms. Both \(6x\) and \(2x\) are considered like terms because they each have the variable \(x\).

To combine them, simply add their coefficients:

• The coefficient of \(6x\) is \(6\).
• The coefficient of \(2x\) is \(2\).

Add these coefficients together: \(6 + 2 = 8\). The result is \(8x\).

Combining like terms is a fundamental part of simplifying expressions, as it keeps expressions concise and easy to handle. By focusing on the coefficients and matching variables, you can simplify complex equations into manageable pieces.

Translating Phrases to Algebra

Translating English phrases into algebra is a crucial skill in math that involves turning words into mathematical symbols and operations. Let’s see how this was achieved in our example.

We started with the phrase 'the difference between the product of six and a number, and negative two times the number.' We need to break this down:

• 'The product of six and a number' translates to \(6x\), which means 6 times the variable \(x\).
• 'Negative two times the number' translates to \(-2x\).
• 'The difference between' indicates subtraction between these two parts.

Putting it all together, the phrase becomes the algebraic expression \(6x - (-2x)\).

This process of translation helps us abstract complex real-world problems into straightforward mathematical expressions. By understanding each component of the phrase, we set the foundation for subsequent operations like simplifying and combining terms. So when translating, always identify the key operations and match them to algebraic equivalents, ensuring each segment of the phrase rightly represents the expression intended.