Problem 125 - College Algebra

Question

Write 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 'The difference between the product of five and a number and twice the number' translates to the algebraic expression '5x - 2x'. After simplifying this expression, you get '3x'. Therefore, '3x' is the final answer.

Step 1: Identify the variable

From the problem, it's stated that the variable to represent the number is \(x\). So when the problem refers to a 'number', use 'x'.

Step 2: Translate the English Phrase into an Algebraic Expression

In maths, 'product' means multiplication and 'difference' means subtraction. The phrase 'the product of five and a number' can be translated into '5x'. The phrase 'twice the number' can be translated as '2x' . The word 'difference' implies subtraction. Therefore, the entire English phrase 'The difference between the product of five and a number and twice the number' can be translated to '5x - 2x'.

Step 3: Simplify the Algebraic Expression

Now, simplify the expression '5x - 2x'. In their order 5x - 2x is equal to '3x'.

Key Concepts

Translating English to Algebra

Turning English phrases into algebraic expressions is a crucial skill in algebra. To do so, it's important to identify keywords that indicate mathematical operations. For example, the word 'product' refers to multiplication, and 'difference' refers to subtraction. Similarly, 'sum' would mean addition, and 'quotient' would refer to division. Once you identify these keywords, the next step is to assign variables to represent unknown numbers, commonly chosen as letters like x, y, or z.

In this exercise, for instance, 'the product of five and a number' translates to the multiplication of 5 and a variable, expressed as 5x. The phrase 'twice the number' means two times the variable, which is 2x. By recognizing these terms, we can form the expression 5x - 2x to represent the English phrase provided.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression as basic or as 'clean' as possible. This involves combining like terms and reducing expressions to their simplest form. Like terms are terms that have the same variables raised to the same power. When simplifying, you combine the coefficients (the numbers in front of the variables) of these like terms while keeping the variable part unchanged.

For our example, the expression 5x - 2x contains like terms because both terms have the variable x without any exponents. Combining the coefficients (5 and -2) gives us 3, and since the variable x remains the same, the simplified expression is 3x. This simplification is key to solving more complex equations efficiently.

Variable Representation

In algebra, variables are symbols that are used to represent unknown values. They are typically represented by letters of the alphabet. The choice of letter is arbitrary, but it's common to use x, y, and z for unknowns in equations. Variables can stand for any number, which allows us to create general solutions to problems.

In the provided exercise, x is chosen to represent 'the number'. This variable becomes a placeholder for any possible value that the number can take. The ability to represent numbers with variables is fundamental in forming algebraic expressions and plays a vital role in all algebraic computations that follow.

Mathematical Operations

Understanding mathematical operations is essential for translating phrases into algebra. These operations include addition, subtraction, multiplication, and division. Algebraic expressions often involve these operations performed on variables and numbers.

For example, in our exercise, the phrase 'the difference between the product of five and a number and twice the number' involves both multiplication ('product of five and a number' being 5x) and subtraction (the 'difference between' the 5x and 2x). Recognizing the operations within English phrases and correctly applying them to variables plays a crucial role in constructing accurate algebraic expressions.