Problem 5 - Prealgebra

Question

Write each of the following English phrases in symbols using the variable \(x\). Five \(x\) decreased by 6

Step-by-Step Solution

Answer

The expression is \( 5x - 6 \).

Step 1: Understanding the Expression

The phrase 'five \( x \)' implies the multiplication of the variable \( x \) by 5, which is written as \( 5x \). We need to translate this part first into symbols.

Step 2: Translating 'Decreased by 6'

The term 'decreased by 6' indicates a subtraction operation where 6 is subtracted from another value, which in this context is \( 5x \).

Step 3: Combining the Symbols

Combine the translation results from the previous steps. \( 5x \) decreased by 6 can be expressed symbolically as \( 5x - 6 \).

Key Concepts

Translating Verbal Phrases to Algebraic Expressions

Translating verbal phrases into algebraic expressions is important for solving real-world problems with math. Imagine language and numbers working together to make complex ideas simple and clear.

When faced with a phrase like "five \( x \) decreased by 6," we are essentially asked to replace the spoken word with mathematical symbols. Here's how we do it:

• For the part "five \( x \)," identify the number attached to \( x \). Here, the number is 5, which means we multiply \( x \) by 5.
• For "decreased by 6," note the word "decreased." It points to a subtraction operation, which indicates taking away a number from another. In our expression, we'll subtract 6 from another term.

At the end, you simply bring these parts together: \( 5x - 6 \). As you practice more, translating phrases to expressions will become second nature.

Variables and Coefficients

Variables and coefficients are the building blocks of algebra. A variable is usually a letter that represents an unknown number, often \( x \), \( y \), or \( z \). Think of it like a container ready to hold any number the problem requires.

A coefficient, meanwhile, is the number that multiplies the variable. If you see \( 5x \), then 5 is your coefficient. This just means we have five times whatever number \( x \) ends up being.

Understanding variables and coefficients:

• Variables let you generalize mathematical problems. They stand in for values that can vary, hence the name "variable."
• Coefficients tell us how many units of the variable we have. They indicate scale or size in expressions.

Grasping both concepts can give you a powerful tool to describe and solve problems efficiently.

Basic Arithmetic Operations

At the heart of solving algebraic expressions are basic arithmetic operations: addition, subtraction, multiplication, and division. These are the tools you use to manipulate expressions and find solutions.

Let's decode each one with friendly explanations:

• Addition (+): Bringing two numbers together to get a bigger number. When translating phrases, this might be shown as "sum" or "increased by."
• Subtraction (-): Taking one number away from another. Words like "decreased by" or "minus" denote subtraction. In our exercise, "decreased by 6" indicates subtracting 6.
• Multiplication (\( \times \)): This is repeated addition. For example, "five times a number" shows multiplication, written as "5\( x \)" for five lots of \( x \).
• Division (÷): Splitting a number into parts. Terms like "divided by" indicate this action, though isn't part of our specific phrase example.

Mastery of these operations makes handling expressions like \( 5x - 6 \) a straightforward process. They provide the fundamental processes to move parts of math from theory to practice.