Problem 54 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$8 y+7+10 y$$

Step-by-Step Solution

Answer

The simplified algebraic expression is \(18y + 7\).

Step 1: Identify like terms

The like terms in this expression are variables of \(y\), which are \(8y\) and \(10y\). We will combine these two terms.

Step 2: Combine like terms

Combine the like terms by adding \(8y\) and \(10y\). The result is \(18y\). After combining, the expression becomes \(18y + 7\).

Key Concepts

Understanding Like Terms in Algebra

In algebra, recognizing like terms is a crucial part of simplifying expressions. Like terms are terms that have identical variable parts. This means they have the same variable raised to the same power. For instance, in the expression \(8y + 10y + 7\), the like terms are \(8y\) and \(10y\), since they both involve the variable \(y\) to the power of 1.

Understanding this concept is simple once you know the key:

• **Variables must match:** Ensure that the variables are the same in each term.
• **Exponent must match:** Check to see if the exponents on these variables are also the same.
• **Constants don't count:** Numbers without variables are not considered like terms.

The number in front of the variable, called the coefficient, does not affect whether terms are like terms. This means \(2x\) and \(5x\) are like terms, but \(2x\) and \(2y\) are not.

The Process of Combining Like Terms

Once you identify like terms, the next step is combining them, which involves simple addition or subtraction of the coefficients. For our example, \(8y + 10y + 7\), the like terms \(8y\) and \(10y\) are combined by adding their coefficients:

• Add the coefficients: \(8 + 10 = 18\).
• Keep the variable the same: The combined term becomes \(18y\).
• Include any constants: The final expression remains as \(18y + 7\).

By carefully adding or subtracting the coefficients, we create a simplified version of the original expression. It's important to do this wherever possible to make algebraic expressions easier to manage and understand. Remember, only like terms can be combined this way.

Exploring Algebraic Expressions

Algebraic expressions can consist of variables, constants, and operators like addition or subtraction. In the expression \(8y + 7 + 10y\), elements include variables like \(y\), constants like 7, and the plus signs which are operators.
It's helpful to remember:

• **Variables:** Symbols like \(y\) represent numbers and can change their value.
• **Constants:** Fixed numbers like 7 that do not change.
• **Operators:** Symbols like + or - allow you to combine variables and constants.

Understanding how to simplify algebraic expressions by identifying and combining like terms helps in making calculations more straightforward and manageable. The goal is to condense the expression to its simplest form, making it clearer and often easier to work with in equations or while evaluating expressions.