Problem 68 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. . Three less than 4 times a number gives 29 .

Step-by-Step Solution

Answer

The sentence 'Three less than 4 times a number gives 29' can be translated into the following mathematical equation: \(4x - 3 = 29\).

Step 1: Understand the problem

The task is to convert a sentence into a mathematical equation. The sentence refers to a certain number represented by a variable \(x\). The sentence to be converted is 'Three less than 4 times a number gives 29'.

Step 2: Translate the sentence into an equation

The sentence 'Three less than 4 times a number gives 29' can be translated into the following equation: \(4x - 3 = 29\), where \(x\) represents the number. The phrase 'Three less than 4 times a number' translates to \(4x - 3\) and 'gives 29' translates to \(= 29\).

Step 3: Write the final equation

Hence, the sentence 'Three less than 4 times a number gives 29' is written as the following equation: \(4x - 3 = 29\)

Key Concepts

Equation Translation

In mathematics, equation translation involves converting verbal statements or word problems into algebraic equations. This process requires identifying mathematical operations within the context of the statement and expressing these using mathematical symbols. For example, the statement "Three less than 4 times a number gives 29" involves the following translations:

• "4 times a number" means multiplying a variable, which we'll denote as \(x\), by 4, resulting in \(4x\).
• "Three less than" indicates a subtraction operation, so "three less than \(4x\)" becomes \(4x - 3\).
• The phrase "gives 29" translates to an equation equal to 29, hence \(= 29\).

Putting it all together, the statement translates to the equation \(4x - 3 = 29\). This translation is a foundational skill for solving algebraic problems, allowing students to bridge the gap between verbal descriptions and mathematical expressions.

Variables in Equations

Variables are symbols used to represent unknown or changeable values in mathematical equations. In the context of an equation, they provide a way to model and solve problems that involve unknown quantities. For the problem at hand, the variable \(x\) is used to represent "a number" whose value we aim to discover.

• The use of variables is crucial because they allow the formulation of general statements about relationships between numbers.
• In our example, \(x\) is pivotal to forming the equation \(4x - 3 = 29\), where it represents an unknown number we are solving for.
• By systematically manipulating equations, mathematicians can uncover solutions, predict outcomes, and model real-world situations.

Understanding the concept of variables helps in recognizing that they are placeholders in equations that can be adjusted or manipulated to solve for specific numerical outcomes.

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. A linear equation is one where the variable is raised to the power of one, and it forms a straight line when graphed. For the equation \(4x - 3 = 29\), solving it involves the following steps:

• Start by isolating the variable term on one side of the equation. Add 3 to both sides to get \(4x = 32\).
• Then, divide both sides by 4 to solve for \(x\), resulting in \(x = 8\).

These steps highlight the fundamental techniques of adding and dividing terms to isolate the variable, a central part of solving equations. The completed equation \(x = 8\) tells us that when \(x\) is 8, the equation \(4x - 3 = 29\) holds true. Mastery in solving linear equations enables students to tackle more complex problems by systematically applying these basic principles.