Problem 40 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. When two is subtracted from some number, the result is ten.

Step-by-Step Solution

Answer

The equation is \( x - 2 = 10 \).

Step 1: Identify the Unknown Number

Let's consider the unknown number as a variable. We'll use \( x \) to represent this unknown number that the problem is asking about.

Step 2: Translate 'Two is Subtracted from Some Number'

The phrase 'two is subtracted from some number' means we are taking 2 away from our unknown number \( x \). This can be represented in a mathematical expression as \( x - 2 \).

Step 3: Translate 'The Result is Ten' into an Equation

The phrase 'the result is ten' indicates that the expression \( x - 2 \) is equal to 10. Therefore, we can write this relationship as the equation \( x - 2 = 10 \).

Key Concepts

Mathematical Translation

Mathematical translation involves converting sentences or phrases into mathematical expressions or equations. This is a crucial skill in algebra because it allows you to take everyday language and transform it into something that can be manipulated mathematically. In the problem given, we start by translating the sentence, "When two is subtracted from some number, the result is ten." To do this, we need to identify keywords in the sentence that correlate to mathematical operations.

• "some number" refers to an unknown value which we can assign a variable to.
• "two is subtracted from" indicates a subtraction operation.
• "the result is" suggests an equation that equals a certain value.

Understanding these components helps in forming an accurate mathematical model of the language provided. It is essential to be precise in this translation process because even a small misinterpretation can change the problem you are solving entirely.

Unknown Variables

In algebra, unknown variables represent numbers that we do not know yet but want to find. We use letters like "x," "y," or "z" to denote these unknowns. In this exercise, we have the phrase "some number," which implies an unknown variable that can be represented as "x."
Assigning letters to these unknowns is not arbitrary. It allows us to perform operations and solve problems systematically. By representing unknowns as variables, you can set up equations that describe mathematical relationships between different pieces of the problem.

• Choose a variable that clearly represents what you are solving for in the problem.
• Ensure that each unknown variable is defined clearly to avoid confusion.
• Remain consistent with the variables you choose to simplify solution steps.

Equation Solving

Once you have translated the problem into a mathematical equation, the next step is to solve it. In our exercise, we have the equation: \[ x - 2 = 10 \] Solving this equation involves finding the value of "x" that makes the equation true. To solve it, you perform operations that will isolate "x" on one side of the equation.
Let's look at the process:

• Add 2 to both sides of the equation to cancel out the subtraction: \[ x - 2 + 2 = 10 + 2 \]
• This simplifies to \[ x = 12 \]
This process shows that the unknown variable "x" has a value of 12 in the original problem. Solving equations often involves simple arithmetic operations like addition or subtraction to both sides, ensuring that the balance of the equation is maintained.