Problem 23 - Prealgebra

Question

Simplify expression. \(5 y+y\)

Step-by-Step Solution

Answer

The simplified expression is \(6y\).

Step 1: Identify Like Terms

In the expression \(5y + y\), the terms \(5y\) and \(y\) are like terms because they both contain the variable \(y\). Like terms can be combined.

Step 2: Combine Like Terms

To combine the like terms \(5y\) and \(y\), think of \(y\) as \(1y\). Add the coefficients of \(y\), which are 5 and 1. Thus, \(5y + 1y = 6y\).

Key Concepts

Understanding Like Terms

To simplify algebraic expressions effectively, you'll often come across a core concept known as "like terms." Like terms are terms in an expression that have the same variable part, meaning both the variable and its exponent are identical. For example, in the expression \(5y + y\), the terms \(5y\) and \(y\) are like terms because they both contain the variable \(y\).

• Variable Match: The variables need to match completely. \(3x^2\) and \(4x^2\) are like terms, but \(3x\) and \(4x^2\) are not.
• Coefficients: The coefficients (numerical part) can be different and are what we'll add or subtract when combining.

Understanding like terms is essential because it allows us to add or subtract terms to simplify expressions further.

Combining Like Terms: The Simplification Key

Once like terms are identified, the next step is to combine them, which is crucial in simplifying algebraic expressions. In simple terms, you add or subtract the coefficients of the like terms. Think of it like adding apples to apples!

• Step-by-Step: In the example \(5y + y\), we treat \(y\) as \(1y\). Then, add the coefficients: 5 (from \(5y\)) plus 1 (from \(1y\)), giving \(6y\).
• No Change in Variables: Remember, only the coefficients change; the variable with its power stays identical.

By mastering this, you'll find that simplifying expressions becomes an intuitive process.

What Are Algebraic Expressions?

Algebraic expressions are mathematical phrases that contain numbers, variables, and operations, but no equal sign. They can represent real-world relationships and can be simplified by using like terms and combining like terms.

• Components: They typically include coefficients, variables, and constants. For example, in \(3x + 5\), the coefficient is 3, the variable is \(x\), and the constant is 5.
• Operations: May involve addition, subtraction, multiplication, and division.

When working with algebraic expressions, the goal is often to simplify or evaluate them, using techniques like combining like terms, to make them easier to understand or solve for variables when part of equations.