Problem 92 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let x represent the number. The difference between 9 times a number and \(-4\) times the number.

Step-by-Step Solution

Answer

The algebraic expression equivalent to the given English phrase, when simplified, is \(13x\).

Step 1: Translate the English Phrase to Mathematical Expression

The problem asks for the difference between 9 times a number and -4 times the number. This can directly be translated into the expression: \(9x - (-4x)\). It's important to note that taking away a negative value is equivalent to adding a positive one.

Step 2: Simplify the Mathematical Expression

Now, simplify the expression. A minus and a minus going together turn into a plus, so the expression simplifies to \(9x + 4x\).

Step 3: Final Simplification

Combine the like terms to get a final simplified expression: \(13x\) as the final output from the simplification process.

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is like tidying up a room; you want everything to be in order and easy to understand. When you simplify an expression, you are essentially condensing it to the simplest form possible. In the problem given, we start with the expression \(9x - (-4x)\).
To simplify, look for opportunities to combine terms or simplify operations. Here, subtracting a negative, \(-4x\), turns into adding a positive: \(9x + 4x\). Remember, two negatives make a positive.
After identifying these, the final expression becomes \(13x\). Simplification makes expressions easier to work with, ensuring clarity and precision in solving algebraic problems.

Mathematical Translation

Mathematical translation involves converting English phrases into algebraic expressions. This skill is essential for solving word problems and understanding the real-world applications of algebra.
In the example, the phrase 'difference between 9 times a number and \(-4\) times the number' must be translated into an expression. Here, "difference" means we're dealing with subtraction, and "9 times a number" and "\(-4\) times a number" are the quantities involved.
Breaking it down:

• "9 times a number" translates to \(9x\) where \(x\) represents the number.
• "-4 times a number" becomes \(-4x\).
• Combining these gives us the expression \(9x - (-4x)\).

Mastering translation helps in interpreting and solving a wide range of mathematical problems.

Like Terms

Like terms are components in algebraic expressions with the same variable raised to the same power. These terms are essential because they allow you to simplify expressions effectively by combining them.
In our exercise, the expression \(9x - (-4x)\) contains like terms: \(9x\) and \(-4x\). Both terms have the variable \(x\). This commonality means we can add or subtract these terms directly.
By recognizing \(9x\) and \(4x\) as like terms, we simplify the expression to \(13x\) by performing the operation \(9x + 4x = 13x\). Remember:

• Like terms must have the exact same variable part. The coefficients (the numbers in front of the variables) can be different.
• Combining like terms is crucial for reducing expressions to their simplest form.

Understanding how to identify and combine like terms is a foundational skill in algebra.