Problem 95 - Algebra A Combined Function

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. Thirteen minus three times a number is \(13 .\)

Step-by-Step Solution

Answer

The equation is \( 13 - 3x = 13 \).

Step 1: Identify the unknown

The sentence 'Thirteen minus three times a number is 13' contains an unknown number. We will represent this unknown number with the variable \( x \).

Step 2: Translate the words into numbers and variables

The phrase 'three times a number' can be represented as \( 3x \).

Step 3: Set up the equation

The sentence states 'Thirteen minus three times a number is 13'. We can translate this into the equation: \( 13 - 3x = 13 \).

Step 4: Write the equation

The equation representing the sentence is \( 13 - 3x = 13 \).

Key Concepts

Variables

In algebra, a variable is essentially a symbol, usually a letter like \(x\), that represents an unknown number or value in an equation or inequality. The purpose of using variables is to express general relationships between numbers in a way that can be easily manipulated to solve problems. When faced with problems involving unknown numbers, identifying and using variables helps greatly simplify and organize the process.
A key step in solving algebraic equations is being able to recognize what is unknown. For instance, in the sentence "Thirteen minus three times a number is 13," the phrase "a number" refers to the unknown quantity. We use the variable \(x\) to stand in place of this unknown. By doing so, we transform the problem into a more familiar mathematical form that can be analyzed and solved using algebraic techniques.

Translating Sentences

Translating sentences into mathematical language is a crucial skill in algebra that allows us to bridge the gap between verbal descriptions and mathematical equations. This involves interpreting words or phrases and representing them by numbers, operations, and variables. Let's break down the sentence "Thirteen minus three times a number is 13" to better understand how this works.

• "Thirteen" directly translates to the number \(13\).
• "Minus" indicates the subtraction operation.
• "Three times a number" can be interpreted as \(3x\), wherein \(x\) is the variable representing the unknown number.
• "Is" implies equality, guiding us to use the equal sign \(=\).

By identifying these elements, we translate the sentence into the algebraic equation \(13 - 3x = 13\). Practicing with various sentences helps sharpen this translation skill, making it easier to form accurate equations and inequalities.

Equations and Inequalities

An equation is a mathematical statement that asserts the equality of two expressions. In our example, \(13 - 3x = 13\), the equation shows that the result of subtracting three times an unknown number from thirteen equals thirteen. Solving equations typically involves finding the value of the variable that makes this statement true.
Inequalities, on the other hand, explain a relationship where expressions are not equal but instead are greater or lesser than one another, indicated by symbols like \(<\), \(>\), \(\leq\), or \(\geq\). While equations have a specific solution, inequalities describe a range of possible values.
Understanding and setting up the right equation or inequality is fundamental to solving algebraic problems. It starts with the translation of verbal sentences and progresses to manipulation of these statements to isolate and solve for variables. The practice of forming and interpreting equations and inequalities enhances logical thinking and problem-solving skills.