Problem 22 - Prealgebra

Question

Simplify expression. \(y+10 y\)

Step-by-Step Solution

Answer

The simplified expression is \(11y\).

Step 1: Identify Like Terms

Examine the expression to identify like terms. In this expression, both terms involve the variable \(y\). The terms are \(y\) and \(10y\).

Step 2: Combine Like Terms

Add the coefficients of the like terms together. The coefficient of the first term \(y\) is \(1\), and the coefficient of the second term \(10y\) is \(10\). Thus, \(1y + 10y = 11y\).

Step 3: Write the Simplified Expression

After combining the like terms, the simplified expression is \(11y\).

Key Concepts

Like Terms

In algebra, one of the foundational concepts is the identification of like terms. Understanding like terms is key when simplifying expressions, just like in the exercise with the expression \(y + 10y\). Like terms are terms that have the exact same variable and the same exponent. Only the coefficients, which are the numbers in front of the variables, can be different. For example:

• In the expression \(2x + 3x\), both terms are like terms because they both contain the variable \(x\).
• For \(4y^2 + 5y^2\), these terms are like terms as they both have the variable \(y\) raised to the same power \(2\).

By identifying like terms, you can simplify expressions by combining them, making them much easier to work with. In the original exercise, \(y\) and \(10y\) are like terms, which allowed us to add their coefficients together.

Coefficients

Coefficients are the numerical part of the term in an algebraic expression. They sit in front of the variable and signify how many times the variable is being counted. Identifying and correctly adding coefficients is crucial in simplifying variable expressions. Consider the expression \(y + 10y\):

• The coefficient of \(y\) is 1, because it is the same as \(1y\).
• The coefficient of \(10y\) is 10.

When simplifying expressions, like when combining \(y\) and \(10y\), you simply add the coefficients together. For example, \(1 + 10 = 11\), so the simplified expression becomes \(11y\). Understanding coefficients helps us efficiently manage and manipulate algebraic expressions by focusing on the numerical relationships between terms.

Variable Expressions

A variable expression is an algebraic expression that contains variables, which are symbols that stand in for unknown values and can change. Variables are typically represented by letters such as \(x\), \(y\), or \(z\). These expressions can be quite complex or relatively simple. In the case of our exercise involving \(y + 10y\), the variable expression involves \(y\) as the variable:

• It allows us to explore relationships between numbers in a flexible way, since \(y\) can represent any number.
• When you simplify a variable expression like \(y + 10y\), you are combining all of the like terms by adding their coefficients.

The result is a simpler expression that is easier to work with and often tells us more about the relationship between the terms. Simplifying variable expressions helps you solve equations more efficiently, making algebra a powerful tool for problem-solving.