Problem 22 - Algebra 1: Concepts and Skills

Question

Match the sentence with its equation. Let x represent the number. A number decreased by 4 is 2. A. \(x-4=2\) B. \(x+2=4\) C. \(\frac{x}{4}=2\) D. \(2 x=4\)

Step-by-Step Solution

Answer

The sentence 'A number decreased by 4 is 2' matches the equation \(x - 4 = 2\). So, the correct option is A.

Step 1: Identify Variables

Let 'x' represent the number. This is the unknown we are trying to solve for.

Step 2: Translate the Sentence into an Equation

The sentence says: 'A number decreased by 4 is 2'. This can be translated into an algebraic equation as 'x - 4 = 2'.

Step 3: Match the Equation to the Options

Now, match this equation with the options provided. The correct option is A: \(x - 4 = 2\)

Key Concepts

Variables

In algebra, a variable is a symbol that stands in for a number that we don't yet know. Variables are most commonly represented by letters, such as \( x \), \( y \), or \( z \). In the context of solving equations, the variable is what we are trying to find by isolating it on one side of the equation. This process helps us uncover the value that makes the equation true.

Using variables is incredibly useful because it allows us to write expressions and equations that can be solved to find unknown values. This flexibility in algebra helps us solve a variety of problems by setting up an equation where the variable is a placeholder for these unknowns.

When dealing with sentences that involve an unknown number, like "a number decreased by 4," we typically assign that unknown number a variable like \( x \). In this way, we can translate verbal descriptions into mathematical expressions.

Algebraic Equations

An algebraic equation is a mathematical statement that asserts the equality of two expressions. It often includes constants, variables, and mathematical operations such as addition, subtraction, multiplication, or division.

For example, in the equation \( x - 4 = 2 \), the left side contains a variable \( x \) that is decreased by 4. The right side is a constant, 2. This equation asserts that when you subtract 4 from the unknown number \( x \), it is equal to 2.

The goal when solving an algebraic equation is to find the value of the variable that makes the equation true. This involves manipulating the equation by performing inverse operations to isolate the variable on one side of the equation. By doing this, we derive the value of the variable, providing a solution to the equation.

Translating Sentences into Equations

Translating sentences into equations is a key skill in algebra that involves converting verbal expressions or statements into mathematical ones. This process connects the language of mathematics to the language of everyday verbal descriptions.

Let's consider the sentence: "A number decreased by 4 is 2." The phrase "a number" indicates an unknown, which we can represent with a variable like \( x \). "Decreased by 4" tells us to subtract 4 from \( x \), while "is 2" signifies an equality expressing that the result of \( x - 4 \) equals 2.

Through this process of translation, we formulate the equation \( x - 4 = 2 \).

• The translation of mathematical phrases such as "increased by," "decreased by," "product of," or "divided by," enables us to construct equations accurately.
• This ability is foundational in solving real-world problems where mathematical reasoning is applied to analyze and solve everyday scenarios.