Problem 59 - Elementary Algebra

Question

For the following problems, translate the following phrases or sentences into mathematical expressions or equations. Eight less than some unknown number is three.

Step-by-Step Solution

Answer

Question: Translate the phrase "eight less than some unknown number is three" into a mathematical equation. Answer: x - 8 = 3

Step 1: Identify the unknown number

Let's use the variable x to represent the unknown number.

Step 2: Analyze the phrase

Here's the phrase: "eight less than some unknown number is three." To create a mathematical equation, we need to express this phrase in terms of the variable x (the unknown number).

Step 3: Translate the phrase into a mathematical equation

The phrase "eight less than some unknown number" can be represented as "x - 8". The phrase "is three" means that the previous expression should be equal to 3. So, the equation can be written as: x - 8 = 3

Key Concepts

Translating Phrases into Equations

Translating phrases into equations is a fundamental skill in elementary algebra. Imagine you have a sentence such as "eight less than some unknown number is three." Our goal is to express this sentence using mathematical symbols and numbers. To do this, we first identify the key components of the sentence and understand their mathematical counterparts.

• "An unknown number" usually refers to a variable, like \(x\).
• "Eight less than" suggests subtraction, meaning we will be subtracting 8 from our unknown number.
• "Is" translates to an equals sign (\(=\)).
• "Three" is the outcome of our equation, a constant number.

Thus, the phrase "eight less than some unknown number is three" becomes the equation \(x - 8 = 3\). It is crucial to take each part of the sentence and carefully translate it into its mathematical form to ensure accuracy.

Unknown Variables

In algebra, unknown variables are placeholders for numbers we do not yet know. They're typically represented by letters of the alphabet, with \(x\), \(y\), and \(z\) being the most commonly used. When a problem mentions an unknown number, this is our signal to use a variable.
The reason we use variables is to formulate a problem that can be solved for these unknown numbers. For instance, in "eight less than some unknown number is three," we denote the unknown number as \(x\). This representation allows us to write a clear equation that we can manipulate mathematically.
Variables simplify complex sentences and situations, making them a powerful tool in solving algebraic equations.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and mathematical operations that represent a specific value or relationship. When we say "eight less than \(x\)," we are describing a mathematical expression, \(x - 8\), which is part of our original phrase.

• A mathematical expression does not include an equals sign (\(=\)), as it is not asserting equality but merely describing a quantity.
• Expressions can be as simple as a single number or involve several terms, such as \(x - 8\).

By understanding the parts of a mathematical expression, students can better translate phrases and form equations that model real-world problems or abstract concepts. Using expressions allows us to manipulate and simplify complex problems into actionable mathematical steps.

Equation Solving

Equation solving is a process aimed at finding the value of the unknown variable that satisfies the equation. After translating a phrase into an equation, such as \(x - 8 = 3\), our task is to determine the value of \(x\) that makes the equation true.

To solve \(x - 8 = 3\), we perform opposite operations. Since 8 is subtracted from \(x\), we do the reverse by adding 8 to both sides of the equation:
\[x - 8 + 8 = 3 + 8 \]
This simplifies to \(x = 11\). Thus, the value of \(x\) that makes the equation true is 11. By solving equations, we find the unknown values that are crucial to understanding and analyzing algebraic problems.