Problem 2 - Intermediate Algebra

Question

Write each sentence using mathematical symbols. See Examples I through 4 and 6 through 8 . The difference of \(y\) and 3 amounts to 12

Step-by-Step Solution

Answer

\(y - 3 = 12\)

Step 1: Identify the Keywords

The statement "The difference of \(y\) and 3 amounts to 12" contains crucial keywords: "difference" and "amounts to". "Difference" refers to subtraction, and "amounts to" indicates equality.

Step 2: Translate into Mathematical Symbols

The difference between two numbers is expressed using subtraction. Here, the two numbers are \(y\) and 3. Therefore, this translates to \(y - 3\). "Amounts to 12" is translated as an equation equal to 12, leading to: \(y - 3 = 12\).

Key Concepts

Mathematical Symbols

Mathematical symbols are like a special language used worldwide to represent mathematical ideas clearly and concisely. They allow us to express complex information quickly, without the need for lengthy explanations. In our example, the phrase: "The difference of \(y\) and 3 amounts to 12", is translated into symbols as \(y - 3 = 12\). Here:

• \(y\) stands for an unknown value (a variable).
• The minus sign (\(-\)) represents subtraction.
• The equals sign (\(=\)) signifies that two expressions are equal.

Each symbol carries specific meaning, and together they form a coherent mathematical statement that can be read and understood universally.

Subtraction

Subtraction is one of the most fundamental operations in arithmetic and algebra. It represents the process of taking away one quantity from another. The result gained from subtraction is called the difference.In our example, the concept of difference is central. When translating the phrase "the difference of \(y\) and 3," we use subtraction to show how much \(y\) is greater than 3. This operation is represented mathematically with the minus sign: \(-\). For example, if \(y\) were 10, then the subtraction \(y - 3\) would be calculated as 10 minus 3, equaling 7.

Translating Statements

Translating statements into equations involves converting words into mathematical expressions. This is a vital skill in algebra, as it helps solve real-world problems using mathematical techniques. To translate, you need to:

• Identify keywords that indicate mathematical operations. In our example, "difference" suggests subtraction, and "amounts to" implies equality.
• Determine what the sentence is asking you to do with the numbers involved.

By understanding these translations, you can bridge the gap between verbal statements and mathematical precision, enabling problem-solving in various contexts.

Algebraic Expressions

An algebraic expression is a mathematical phrase involving numbers, variables, and operation symbols. It doesn't have an equal sign, but an equation does.In the given exercise, \(y - 3\) is an expression representing the difference between the unknown number \(y\) and 3. When we add "= 12", it becomes an equation, which you can solve to find the value of \(y\).Expressions allow us to generalize mathematical relationships, making it easier to apply rules and solve problems without knowing the value of every number involved. Understanding algebraic expressions is key to mastering algebra and tackling more advanced math topics.