91Ó°ÊÓ

In algebra, variables are symbols that represent unknown or changeable values. These symbols are usually letters like \(x\), \(y\), or \(t\). Variables are essential as they allow for the generalization of mathematical equations and expressions. Instead of working with specific numbers, variables provide the flexibility to work with different scenarios.
For example, in our exercise, the variable \(t\) does not have a specified value. It represents an unknown or variable quantity. Understanding variables is crucial because they are the foundation for building algebraic expressions. They act as placeholders that help us solve problems in both simple and complex mathematical contexts. By using variables, we can describe patterns, create formulas, and solve equations efficiently.

Arithmetic operations are the basic mathematical processes used to calculate values. The four fundamental operations are addition, subtraction, multiplication, and division. Each operation has its unique symbol and constructs relationships within numbers and variables.
In our exercise, the phrase "increased by" indicates the use of addition. Addition is symbolized by the plus sign \(+\) and involves combining two or more quantities into a larger one. For example, if we add 3 and 5, we get 8, which means we increased 3 by 5. This same principle applies when dealing with variables. If \(t\) is our starting value, then \(t + 10\) represents the variable \(t\) having been increased by 10. Understanding these operations is crucial for translating verbal phrases into algebraic expressions, which we will explore next.

Translating verbal phrases into algebra is an important skill in mathematics. It involves converting everyday language into mathematical expressions. This skill helps bridge the gap between real-world problems and mathematical solutions. Let’s look at how this applies to our specific example.
In the phrase "\(t\) increased by 10," we begin by identifying our variable, which is \(t\), and our operation, which "increased by" suggests. In mathematics, "increased by" is linked to addition. Therefore, we take our variable \(t\) and add the constant 10 to it. This gives us the algebraic expression \(t + 10\).

• First, identify key terms: variables like \(t\) and constants like 10.
• Next, recognize the operation suggested by words such as "increased," "decreased," "times," or "divided by."
• Finally, write a new expression that combines the variable and operations accurately.

Mastering this translation allows us to take real-world situations, express them mathematically, and solve them using algebraic methods.