Problem 74 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. Four times a number is equal to 25 decreased by the number.

Step-by-Step Solution

Answer

\[4x = 25 - x\].

Step 1 - Identify the Unknown

In this sentence, it is mentioned that \(x\) represents the number. This is our unknown variable.

Step 2 - Translate the sentence into mathematical terms

'Four times a number' can be translated into \(4x\) in mathematical terms. 'Equal to' corresponds to the '=' sign. '25 decreased by the number' is represented by \(25-x\). Putting it all together, we get the following equation: \(4x = 25 - x\).

Key Concepts

Variable Representation

Understanding variable representation is key in solving algebraic equations. In mathematics, a variable acts as a placeholder for an unknown value. This allows us to generalize mathematical problems and solve them systematically.
In the exercise, we are asked to let the variable \(x\) represent the unknown number. By clearly defining \(x\), we simplify the expression and can articulate the problem more effectively. For any situation like this, keep in mind:

• A variable is typically denoted by letters, with \(x\) being most commonly used, but you can choose any other letter if preferred.
• Consistently use the chosen variable throughout the problem to avoid confusion.

By thinking of \(x\) as the number we wish to find, we can proceed to translate word problems into solvable mathematical equations easily.

Translation from Words to Math

Converting a verbal statement into a mathematical equation is an essential skill in algebra. Once we understand the problem, we need to express it using mathematical symbols. This process might seem complex at first, but with practice, it becomes more intuitive.
In our exercise, the phrase "Four times a number" was translated to \(4x\). Here, the word 'times' indicates multiplication. Meanwhile, "is equal to" is directly translated to the '=' sign in math. The statement "25 decreased by the number" was expressed as \(25 - x\), where 'decreased by' corresponds to subtraction. Some helpful pointers include:

• Identify key phrases that relate to mathematical operations: 'times' for multiplication, 'decreased by' for subtraction, 'is equal to' for equality, etc.
• Be aware that every part of the sentence might be represented by a mathematical symbol or operation.

By mastering this translation skill, you can convert real-world problems into equations that are ready for solving.

Solving Equations

Once an equation is established, solving it involves finding the value of the variable that makes the equation true. This process requires strategic techniques to simplify and isolate the variable on one side of the equation.
In the example equation \(4x = 25 - x\), we aim to solve for \(x\). Here’s a quick rundown on how to tackle similar equations:

• First, eliminate any variables on one side by either adding or subtracting them from both sides of the equation. For our equation, add \(x\) to both sides to get: \(4x + x = 25\).
• Combine like terms to simplify: \(5x = 25\).
• Finally, solve for \(x\) by isolating it. Divide both sides by 5: \(x = 5\).
• Check your solution by substituting the value back into the original equation to ensure both sides equal.

Solving equations may involve different operations, so stay mindful of the steps needed to maintain balance and correctness throughout the solution.