Problem 41 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. When four is subtracted from some number, the result is thirty-one.

Step-by-Step Solution

Answer

The equation is \( x - 4 = 31 \).

Step 1: Identify the unknown number

We denote the unknown number by a variable, typically using the letter \( x \) to represent it. This helps us create a mathematical expression.

Step 2: Formulate the subtraction expression

According to the problem, we need to subtract "four" from the "some number." So, the subtraction expression will be \( x - 4 \).

Step 3: Set the expression equal to the given result

The problem states that when this subtraction is performed, the result is "thirty-one." Therefore, we write the equation as \( x - 4 = 31 \).

Key Concepts

Understanding Variables

In algebra, a variable is essentially a symbol that stands for a number we do not know yet. We often use letters like \( x \), \( y \), or \( z \).

• Variables allow us to create expressions and equations that help solve problems where the exact numbers are unknown or can change.
• They can be placeholders for a single value or represent a range of values.

By assigning a variable like \( x \), we can easily manipulate it within various mathematical operations like addition, subtraction, and more complex calculations. In the example where "some number" is unknown to us, we assign it the variable \( x \). This makes it easier to organize and solve the problem step-by-step. Variables are the backbone of algebraic expressions, making them a fundamental tool in problem-solving and allowing for flexibility in mathematical modeling.

Building and Solving Equations

An equation is a mathematical statement that shows the equality between two expressions. In the problem provided, we created an equation using subtraction: \( x - 4 = 31 \).

• An equation consists of two sides separated by an equal sign (\( = \)).
• Both sides must be balanced, meaning they represent the same value for the equation to be true.

To solve an equation, we usually perform a series of steps to isolate the variable on one side, determining its value. In this example, solving the equation \( x - 4 = 31 \) involves adding 4 to both sides to isolate \( x \). This is a basic form of solving equations, but as we delve deeper into algebra, equations can become more complex involving multiple variables and operations.

Applying Arithmetic Operations

Arithmetic operations are basic computations we perform with numbers: addition, subtraction, multiplication, and division. These operations are vital in forming and solving equations, like in our example sentence.

• Subtraction is used when we remove a value from another, such as subtracting 4 from a number.
• All arithmetic operations follow specific rules and order, known as the order of operations (PEMDAS/BODMAS).

In our exercise, the key arithmetic operation is subtraction. "When four is subtracted from some number," we express this in mathematical form as \( x - 4 \). Understanding these operations helps to break down problems into manageable parts, allowing us to focus on each step separately. Arithmetic provides the building blocks needed for more advanced algebraic concepts.