Problem 45 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. When one is subtracted from three times a number, the result is eight less than six times the original number.

Step-by-Step Solution

Answer

Equation: \( 3x - 1 = 6x - 8 \)

Step 1: Identify the unknown

Let's assign a variable to the unknown number we are trying to find. We'll use \( x \) to represent this number.

Step 2: Translate "three times a number"

The expression "three times a number" can be written as \( 3x \).

Step 3: Translate "one is subtracted from three times a number"

The phrase "one is subtracted from three times a number" can be written as \( 3x - 1 \).

Step 4: Translate "eight less than six times the original number"

The expression "six times the original number" is \( 6x \). To get "eight less than" this amount, we write it as \( 6x - 8 \).

Step 5: Form the equation

Since the problem states that "the result is," we set the two expressions equal to each other. Thus, we have the equation: \( 3x - 1 = 6x - 8 \).

Key Concepts

Unknown Variable

When approaching a mathematical problem that involves an unknown quantity, it is crucial to first identify the unknown variable. This is typically represented by a letter, most commonly, by tradition, the letter \( x \). This allows us to label the unknown in a way that we can easily reference and manipulate in various mathematical operations. Simply put, an unknown variable acts as a placeholder for a value we are yet to determine.

Choosing a variable is the first step in unraveling a word problem. For example, if you are trying to find an unknown number in a given scenario, you might say: "Let \( x \) be the number we are looking for."

With this representation, every time the problem references this unknown number, we can simply use \( x \). This makes our calculations and the formation of mathematical expressions much more streamlined.

Translating Phrases

Turning phrases from a word problem into mathematical expressions is a fundamental skill in algebra. It involves understanding and identifying key terms and their mathematical counterparts.

Here are some common translations:

• "Three times a number" translates to \( 3x \) if the number is represented by \( x \).
• "Subtracted from" means you will subtract a quantity from another, like in \( 3x - 1 \), which reads as "one is subtracted from three times a number."
• "Less than" often reverses the order of terms. For example, "eight less than six times a number" becomes \( 6x - 8 \), where you subtract 8 from \( 6x \).

Using these simple translation rules, complex word problems can be converted into understandable, solvable algebraic expressions.

Forming Equations

Once phrases are translated into mathematical expressions, forming equations becomes the next logical step. An equation is formed when two expressions are set equal to each other, often identified by phrases such as "is," "equals," or "results in."

In our exercise, after translating both sides of the given problem, we end up with two expressions: \( 3x - 1 \) and \( 6x - 8 \). These two expressions are actually describing the same quantity, allowing us to form the equation \( 3x - 1 = 6x - 8 \).

Forming the equation is pivotal because it sets the stage for solving for the unknown variable. Once the equation is set up, algebraic techniques can be employed to isolate and determine the value of that variable, leading us straight to the solution of the problem.