Problem 56 - Prealgebra

Question

Simplify each expression. $$2 y+6+5 y$$

Step-by-Step Solution

Answer

The simplified expression is \(7y + 6\).

Step 1: Identify Like Terms

To simplify the expression, first identify the like terms. In this exercise, the like terms involving the variable are \(2y\) and \(5y\), and the constant term is \(6\).

Step 2: Combine Like Terms

Add the coefficients of the like terms. The terms \(2y\) and \(5y\) have the same variable, so we sum their coefficients: \(2 + 5 = 7\). Thus, we have \(7y\).

Step 3: Write the Simplified Expression

Combine the simplified term \(7y\) with the constant term \(6\) that has no like term to combine with. The final simplified expression is \(7y + 6\).

Key Concepts

Like Terms

In algebra, recognizing and working with like terms is a crucial step in simplifying expressions. Like terms are terms that have exactly the same variable parts. This means they involve the same variable raised to the same power. For instance, in the expression given, both \(2y\) and \(5y\) are like terms because they each have the variable \(y\).To identify like terms easily, look for:

• Terms with the exact same variable.
• Terms with the same exponent on those variables.
• Constant terms (numbers without variables) that can be paired together.

Identifying like terms allows you to combine them, which is an essential step towards simplifying an expression fully.

Coefficients

When we talk about coefficients in algebra, we refer to the numerical part of a term that is multiplied by the variable. In any given expression, like \(2y + 5y + 6\), each term involving a variable will have a coefficient. Here, \(2\) and \(5\) are the coefficients for \(2y\) and \(5y\) respectively.Understanding coefficients helps:

• Easily add or subtract terms when simplifying expressions.
• Identify how much one part of an expression is scaled by a variable.

When combining like terms, we sum their coefficients. For instance, \(2y + 5y\) becomes \(7y\) when we add \(2\) and \(5\). This ability to manipulate coefficients is a fundamental skill in algebraic operations.

Simplified Expressions

A simplified expression is one in which all possible operations have been performed, and it can no longer be simplified further. Simplifying makes expressions easier to understand and work with, especially when solving equations or exploring functions.In the given example, the expression \(2y + 6 + 5y\) simplifies to \(7y + 6\). Here's how it is done:

• Combine all like terms by adding their coefficients, transforming \(2y + 5y\) to \(7y\).
• Identify any constants or other terms that cannot be combined with the like terms, such as the \(6\) in this case.

The result is a much cleaner and more manageable form of the original expression. Simplification not only makes an expression clear and concise but also often reveals important mathematical relationships.