Problem 26 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation or an inequality. Let x represent the number. 18 decreased by a number is 6.

Step-by-Step Solution

Answer

The equation derived from the sentence is \(18 - x = 6\).

Step 1: Identify the variable

Identify the variable, x, which represents the number to be found. The sentence says 'a number',which we'll represent it by \(x\).

Step 2: Understanding the operation ‘decreased by’

In the context of the given sentence, the phrase 'decreased by' indicates subtraction. Therefore, construct the equation accordingly.

Step 3: Convert the sentence to an equation

Now, write the sentence as an equation: \(18 - x = 6\). This should read as '18 decreased by x is equal to 6.'

Key Concepts

Algebraic Expressions

Algebraic expressions are a foundational concept in mathematics, involving numbers, variables, and operations. They serve as a way to represent real-world problems mathematically. Understanding algebraic expressions allows you to manipulate and solve equations. In the context of a problem, we transform verbal sentences into mathematical statements.
For example, when the problem states, "18 decreased by a number is 6," we form the algebraic expression by translating the words into operations with numbers and variables.

• Number: Recognizable constants like 18.
• Variable: An unknown value, often represented by letters like x.
• Operations: Mathematical instructions like addition and subtraction.

Combining these elements, we create an equation: "18 - x = 6." Mastering this translation is key to solving more complex mathematical problems.

Variable Identification

Variable identification is an essential skill in writing equations. Variables act as placeholders for unknown values in algebraic expressions or equations. Identifying the variable means selecting a symbol, usually a letter, that stands in for something we need to find.
In our exercise, the phrase "a number" is where we need to identify a variable. "A number" suggests an unknown value, which is represented as the variable 'x'. This step is crucial as it forms the basis of our equation.

• Choose a letter: Select a letter, commonly x, y, or z, to represent the unknown value.
• Consistent usage: Use the same variable throughout the equation to avoid confusion.
• Connection to the problem: Relate the variable back to the described component in the problem, ensuring proper understanding.

By consistently identifying variables accurately, solving algebraic equations becomes more structured and less ambiguous.

Subtraction in Equations

Subtraction is a common operation in equations, often denoted by phrases like "decreased by". Grasping its application allows us to effectively model real-world scenarios. In this exercise, the phrase "decreased by" signifies the use of subtraction in forming our equation.
Understanding the role of subtraction in equations helps in not only setting up but also solving them correctly.

• Identify the operation: Recognize words or phrases indicating subtraction, like "less" or "decreased by".
• Placement in expression: Subtraction typically follows the order described, such as "a minus b," represented as \(a - b\).
• Solve for the variable: Follow through by using inverse operations, such as adding the same number to both sides of the equation for clarity and solution.

In our case, "18 decreased by x is 6" gets translated directly into the equation: \(18 - x = 6\). Recognizing the role of subtraction here is instrumental to developing a correct solution.