Problem 26 - Prealgebra

Question

Simplify expression. \(2 y+8+5 y+1\)

Step-by-Step Solution

Answer

The simplified expression is \(7y + 9\).

Step 1: Identify Like Terms

The expression given is \(2y + 8 + 5y + 1\). First, identify the terms that are alike. In this case, \(2y\) and \(5y\) are like terms since they both contain the variable \(y\). Similarly, \(8\) and \(1\) are like terms as they are both constants.

Step 2: Combine Like Terms with Variable

Combine the like terms with the variable \(y\). Add \(2y\) and \(5y\) together to get \(7y\).

Step 3: Combine Constant Terms

Now, add the constant terms together. Add \(8\) and \(1\) to get \(9\).

Step 4: Write the Simplified Expression

Combine the results from Step 2 and Step 3 to write the final simplified expression, which is \(7y + 9\).

Key Concepts

Like Terms

Like terms are terms that have the same variables raised to the same powers. In mathematical expressions, identifying like terms is the first step in simplification. For example, in the expression \(2y + 8 + 5y + 1\), the terms \(2y\) and \(5y\) are considered like terms because they both have the variable \(y\). The constant numbers \(8\) and \(1\) are also like terms because they are plain numbers without variables.

Recognizing like terms helps to efficiently simplify expressions by grouping them together. Remember, terms can only be alike if they have identical variable parts. This rule allows us to transform complex expressions into simpler forms by combining these like terms.

Combining Like Terms

Combining like terms is an essential skill for simplifying algebraic expressions. After identifying like terms, you combine them by performing operations such as addition or subtraction. For instance, when simplifying \(2y + 8 + 5y + 1\), you would add the coefficients of the like terms with variable \(y\), which are \(2y\) and \(5y\).

When you add them, \(2 + 5\) gives \(7\), so you write \(7y\). Similarly, for constant terms \(8\) and \(1\), you simply add them to get \(9\). The result from combining these like terms is a more concise expression \(7y + 9\).

• Always perform operations inside the same category of terms.
• Be careful with signs; if you need to subtract, don't forget the negative signs!

This step is crucial because it simplifies the expression to its most straightforward form.

Variables and Constants

Variables and constants are fundamental components of algebraic expressions. A variable represents an unknown value and is usually denoted by a letter, like \(y\) in our case. Variables can change, which is why they are "variable," meaning they can represent different numbers.

Constants, on the other hand, are fixed values. They do not change, regardless of the rest of the expression. In \(2y + 8 + 5y + 1\), both \(8\) and \(1\) are constants. Understanding the difference between variables and constants in expressions helps guide the simplification process by keeping variable-related terms separate from constant terms.

• Variables are flexible; their values depend on the equation.
• Constants are consistent; their values remain the same.

By identifying variable terms and constant terms, you can combine them efficiently, which is vital in simplifying any expression.