Problem 71 - Elementary and Intermediate Algebra

Question

Simplify by combining like terms. $$ 36 y+y-9 y $$

Step-by-Step Solution

Answer

The simplified expression is \( 28y \).

Step 1: Identify Like Terms

First, observe the given expression: \[ 36y + y - 9y \] Identify the like terms, which are terms with the same variable. In this case, all terms contain the variable \( y \).

Step 2: Group Like Terms

Next, group these like terms together. This allows us to combine them more easily. The expression can be rewritten as: \[ (36y + y - 9y) \].

Step 3: Simplify by Addition and Subtraction

Now, add and subtract the coefficients of the like terms: 1. Notice that \( y \) is the same as \( 1y \).2. Combine the coefficients: - Calculate \( 36 + 1 = 37 \). - Subtract 9 from 37: \( 37 - 9 = 28 \). 3. The expression simplifies to \( 28y \).

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They represent quantities in a general form without giving them exact values. For example, the expression \( 36y + y - 9y \) is an algebraic expression. In this expression:

• \(36y\), \(y\), and \(-9y\) are terms.
• Each term consists of a coefficient and a variable.
• The operations involved are addition and subtraction.

Understanding algebraic expressions is crucial for solving equations and simplifying expressions. It allows us to handle complex mathematical problems more systematically.

Simplification

Simplification helps make algebraic expressions easier to work with by combining like terms or reducing them to their simplest form. The goal of simplification is to represent the expression in the most compact way possible while maintaining its value. In our example of the expression \( 36y + y - 9y \), simplification involves:

• Identifying like terms that have the same variable.
• Grouping these like terms together.
• Adding or subtracting their coefficients to get a single term.

After simplification, the expression \( 36y + y - 9y \) becomes \( 28y \), which is much simpler to work with.

Coefficients

Coefficients are numbers that are multiplied by variables in algebraic expressions. They indicate how many times to add the variable to itself. In the expression \( 36y + y - 9y \):

• The coefficient of \(36y\) is 36.
• Although \(y\) appears to lack a coefficient, it implicitly has a coefficient of 1. This is because \(y\) is the same as \(1y\).
• The coefficient of \(-9y\) is -9, meaning it is to be subtracted.

When simplifying, we combine coefficients of like terms by adding or subtracting them, following arithmetic rules. This step helps in reducing the expression to its simplest form.

Variables

Variables act as symbols that represent unknown quantities in algebraic expressions. They can take various numerical values, which makes them powerful tools for solving equations and modeling real-world situations. In \( 36y + y - 9y \):

• \(y\) is the variable shared by all terms.
• It allows us to focus on the coefficients during simplification because the variable itself is consistent across terms.

By treating the variable as a constant symbol while simplifying, we can easily combine like terms and focus on changing only the coefficients. This makes algebra flexible and useful for diverse applications.