Problem 68 - Introductory Algebra for College Students

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 written as the equation \(4x - 3 = 29\).

Step 1: Identify the given variable and number

Here, the given variable is \(x\) which represents the number and the given number is \(29\) which is the result.

Step 2: Decode the sentence

The phrase '4 times a number' suggests four multiplied by the variable \(x\), written as \(4x\). The phrase 'Three less than' signifies subtracting three from \(4x\), written as \(4x - 3\). 'Gives 29' indicates that this is equal to \(29\).

Step 3: Write the final equation

Piecing all parts together, three less than four times a number giving 29 can be written as \(4x - 3 = 29\).

Key Concepts

Translating Sentences to Equations

Translating sentences into equations is a fundamental skill in algebra that involves identifying mathematical expressions within everyday language. The primary objective is to analyze the statement and represent it accurately using mathematical symbols.
For example, in the given exercise, the sentence "Three less than four times a number gives 29" needs to be logically broken down:

• "Four times a number" indicates multiplying the variable by four, hence it becomes \(4x\).
• "Three less than" suggests subtracting three from the previous expression, leading to \(4x - 3\).
• "Gives 29" implies equality with the number 29, forming the equation \(4x - 3 = 29\).

Translating words into equations is like decoding a language. It requires careful identification of components such as operations, numbers, and variables to ensure an appropriate equation.

Variable Expressions

Understanding variable expressions is crucial in algebra, as they are expressions that include variables, numbers, and operations.
In the original exercise, the term "four times a number" refers to a variable expression. Here, the variable is represented as \(x\), which stands for an unknown number, and the phrase "four times a number" thus becomes \(4x\).
This type of expression tells us that the number \(x\) is unidentified but multiplied by four in this context. Recognizing and creating these expressions from words is essential for solving algebraic problems, as they are the starting point for forming full equations.

Mathematical Operations

Mathematical operations like addition, subtraction, multiplication, and division are key components in forming equations from sentences. Identifying these operations within word problems is necessary to successfully convert them into algebraic equations.
In the example provided, the phrase "three less than" indicates subtraction. The operation here takes place between the result \(4x\) and the number three, resulting in the expression \(4x - 3\).
Recognizing these operations helps in formulating equations correctly. Each word or phrase in a problem corresponds to specific mathematical operations, which are essential in building a precise equation that reflects the original statement.