Problem 73 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. Five times a number is equal to 24 decreased by the number.

Step-by-Step Solution

Answer

The equation corresponding to the given sentence is \(5x = 24 - x\).

Step 1: Identify the variable

The problem specifies that the variable \(x\) should represent the number. Whenever you see 'a number' in the sentence, you should think of \(x\).

Step 2: Translate 'five times a number'

The phrase 'five times a number' can be translated into mathematical terms as \(5x\), meaning five multiplied by \(x\).

Step 3: Translate '24 decreased by the number'

The phrase '24 decreased by the number' translates mathematically to \(24 - x\). This means subtract the number \(x\) from 24.

Step 4: Formulate the equation

The sentence states that 'five times a number is equal to 24 decreased by the number'. In mathematical terms, this can be written as \(5x = 24 - x\). This represents the equation derived from the sentence.

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Variables are symbols that stand for unspecified numbers, and they are often represented by letters like 'x' or 'y'. To understand algebraic expressions, you’ll need to recognize the terms and operations they consist of.

For instance, take the phrase 'five times a number'. In algebraic terms, this is expressed as '5x', where '5' is the coefficient, meaning it's the number we multiply by the variable 'x'. Recognizing these expressions is essential as it builds a foundation for developing more complex mathematical concepts. When studying algebraic expressions, remember these points:

• 'Five times' translates to multiplication.
• 'A number' refers to a variable which we often denote as 'x'.
• Combining these two, we get the expression '5x' to represent 'five times a number'.

By mastering algebraic expressions, you're equipping yourself with the skills to tackle a wide range of mathematical problems.

Equation Formulation

The process of equation formulation involves translating words or real-world situations into mathematical equations. It's a vital skill that allows you to solve problems by creating equations that can be analyzed mathematically. Essentially, you're turning a sentence, with all its quantitative and relational information, into an equation that equates two expressions.

To form an equation from the given problem, 'five times a number is equal to 24 decreased by the number', we do the following:

• Translate 'five times a number' to an algebraic expression as '5x'.
• Understand that 'is equal to' signals the equality, represented by '='.
• Translate '24 decreased by the number' to '24 - x'.
• Combine these components into one equation as '5x = 24 - x'.

This equation reflects the balance between two expressions, and solving it will uncover the value of the variable 'x'. Formulating equations correctly is the first step towards finding solutions to mathematical and real-world problems alike.

Variable Representation

In the realm of algebra, variable representation is the use of a symbol, commonly a letter, to stand in for an unknown quantity in an equation or expression. The choice of symbol can be arbitrary, but it's chosen to make the equations or expressions easier to work with. In our exercise, 'x' is used to represent 'a number' which is unknown.

Variables are incredibly useful because they allow you to work with an unknown number as if it were known. You can perform operations on it, combine it with other numbers or variables, and eventually solve for it. Here are some key insights about variable representation:

• A variable can represent any number and is not restricted to a specific value until it is solved in an equation.
• Variables make it possible to generalize and create formulas that are widely applicable.
• Understanding the representation of variables is essential for translating written problems into solvable equations.

As seen in our example, 'x' can be manipulated just like any other number in an algebraic expression. By maintaining consistency with our variable representation, we provide a pathway to solving complex problems simply and effectively.