Problem 23 - Elementary Algebra

Question

Simplify. $$ 1 y-2 $$

Step-by-Step Solution

Answer

The simplified expression is \(y - 2\).

Step 1: Understand the Expression

The given expression is \(1y - 2\). This expression is a linear expression with a single variable \(y\). Remember, \(1y\) is equivalent to just \(y\), as multiplying any number by 1 gives the number itself.

Step 2: Simplify the Expression

Since \(1y\) is equal to \(y\), we can rewrite the expression \(1y - 2\) as \(y - 2\). Thus, the expression has been simplified to \(y - 2\), which is already in its simplest form.

Key Concepts

Linear Expression

A linear expression is a type of algebraic expression that represents a straight line when plotted on a graph. It's characterized by having no exponents higher than 1. In the example of the expression \(1y - 2\), we can see it's a simple linear expression. The term "linear" comes from the term "line", indicating no squared terms or higher powers. Linear expressions can:

• Include terms with coefficients (like the 1 in \(1y \)).
• Include constants (such as -2 in our expression).
• Be rewritten or rearranged but always maintain a degree of 1 for the variable.

Recognizing linear expressions is crucial in algebra as it forms the basis for solving equations, graphing, and understanding more complex expressions.

Simplifying Expressions

Simplifying expressions is a key process in algebra, aiming to make complex expressions easier to work with by reducing them to their simplest form. The goal is to declutter expressions to make solving and understanding them as straightforward as possible. In our exercise:\[\text{Original expression: } 1y - 2\]Simply by recognizing that \(1y\) is just \(y\), we simplify it to \(y - 2\). This shows that:

• Like terms can be combined or rewritten for simplicity.
• Constants or coefficients can help reshape expressions but don't change inherent values (just like \(1\cdot y = y\)).
• Once simplified, further operations like addition, subtraction, and substitution become much clearer.

This process of simplifying expressions is fundamental as it helps students focus on the core components without unnecessary complexity.

Variables

Variables, like \(y\) in our simplified expression \(y - 2\), are essential components of algebra. They act as placeholders for unknown values and can represent numbers that vary or change. Here are important attributes of variables:

• They allow flexibility to work with equations until specific values are assigned.
• Variables usually are letters from the alphabet, commonly \(x\), \(y\), \(z\), etc.
• In expressions like ours, variables allow students to perform algebraic operations and solve equations for the unknown value.

Understanding variables is crucial as they are foundational for manipulating and solving algebraic equations across all levels of math.

Coefficients

Coefficients are the numerical part of terms in algebra. In \(1y\), the number \(1\) is the coefficient of \(y\). They scale the variable parts of an expression and are crucial in reshaping and simplifying expressions. Let's see some characteristics:

• They can be positive or negative, affecting the direction and positioning of the line in graphing linear expressions.
• Multiplying a variable by a coefficient (like \(1 \times y\)) often simplifies expressions by being reduced to the variable itself if the coefficient is one.
• Identifying and working with coefficients allow students to understand the role each part of an expression plays, especially in complex problem-solving.

Grasping how coefficients influence expressions aids in mastering algebraic manipulation and simplifies broader learning in math.