Problem 125 - 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 five and a number and twice the number.

Step-by-Step Solution

Answer

The English phrase translates to the algebraic expression \(3x\).

Step 1: Identify the mathematical operations

In this phrase, 'product' signifies multiplication, 'twice' signifies multiplying by 2, and 'difference' signifies subtraction. 'Five and a number' corresponds to '5 times x', and 'twice the number' refers to '2 times x' or '2x'.

Step 2: Write the initial algebraic expression

The phrase 'The difference between the product of five and a number and twice the number', can now be written as '5x - 2x'.

Step 3: Simplify the algebraic expression

The expression 5x - 2x combines two like terms and can be simplified to 3x.

Key Concepts

Translating English to Algebra

Understanding algebra involves turning real-world scenarios into mathematical statements, quite like translating one language to another. Let's say you're reading a phrase that describes a relationship or calculation involving a number. Your task is to express that relationship using algebraic symbols.
For example, when you encounter a phrase like 'five times a number', you would translate it into the algebraic expression '5x', assuming 'x' represents that number. Other phrases to look out for are terms like 'the sum of', which points to addition, 'the product of', signaling multiplication, and so on. In the exercise, the phrase translates to '5x' for the product of five and a number, and '2x' for twice the number. By aligning these terms with their algebraic counterparts, we bridge the gap between verbal descriptions and algebraic expressions.

Simplifying Expressions

Once we've translated English phrases into algebraic expressions, we often end up with a mathematical phrase that can be made simpler. Simplifying an expression means to restate it in its most basic form. This involves combining like terms, reducing fractions, or applying other algebraic rules.
In our exercise, we have the expression '5x - 2x'. Here, '5x' and '2x' are like terms because they both represent multiples of 'x'. To simplify, we combine these like terms by subtracting the coefficients (the numbers in front of 'x'). Thus, '5x - 2x' simplifies to '3x'. This condensed expression is easier to work with in equations and offers a clearer picture of the relationship it describes.

Why Simplify?

Simplifying isn't just about making the expression shorter; it's about clarity and ease. A simplified expression removes unnecessary complexity, allowing you to solve problems more effectively and understand the underlying concepts more intuitively.

Algebraic Operations

Algebra is filled with operations that tell us what to do with numbers and variables. The basic algebraic operations include addition, subtraction, multiplication, and division, which can be directly translated from phrases used in English.
Consider the phrase from the exercise that says 'the difference between'. In algebra, 'difference' means subtraction. The operation needed here is to subtract '2x' from '5x'. When you perform this operation, you are actively manipulating the expression to reflect the action described in English.
Knowing your algebraic operations and understanding how they correspond to verbal cues is crucial. It allows you to construct accurate mathematical representations of problems, which in turn sets you up for success in solving them.