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Chapter 2: Problem 27

Simplify each expression, when possible. SEE EXAMPLE 1. (OB]ECTIVE 1) $$9 x+3 y$$

Short Answer

Expert verified
The simplified form of the expression \(9x + 3y\) remains \(9x + 3y\) as it cannot be further simplified.

Step by step solution

01

Identify the term

The term given is \(9x + 3y\).
02

Attempt to Simplify

Try to simplify the term. In this case, the equation cannot be further simplified as the terms are not like terms. Like terms refer to terms that have the same variables and powers. Here, one term contains the variable x, and the other contains the variable y.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Like Terms
Like terms are essential in simplifying algebraic expressions because they allow us to combine terms more efficiently. For terms to be considered "like," they must have the same variable(s) and the same exponent on those variables. This is crucial in recognizing what you can combine together when dealing with algebraic terms.
  • For example, in an expression like \(4x + 2x\), both terms are like terms because they share the variable \(x\) and it is raised to the same power (in this case, \(x^1\)). Thus, they can be combined to \(6x\).
  • However, in the expression \(9x + 3y\), as given in the exercise, the terms are not like because one term is \(x\) based while the other is \(y\) based. Since they contain different variables, they cannot be combined into a single term.
Identifying like terms helps in proper simplification and ensures calculations are made accurately. Always check the variables and their exponents to determine if terms are "like." This understanding is vital for organizing and simplifying longer equations or expressions.
Algebraic Terms
An algebraic term is a building block of algebraic expressions. It consists of coefficients, variables, and exponents. Understanding its structure is fundamental to correctly simplifying and manipulating expressions.Within an algebraic term:
  • A coefficient is a number that multiplies a variable. For example, in \(9x\), 9 is the coefficient.
  • A variable is a symbol that represents an unknown value, commonly \(x\), \(y\), or \(z\).
  • An exponent, if present, shows how many times the variable is multiplied by itself.
Let鈥檚 focus again on the given expression, \(9x + 3y\). Here:
  • "9x" is an algebraic term where 9 is the coefficient, \(x\) is the variable, and the exponent of 1 is implied.
  • "3y" is another algebraic term where 3 is the coefficient, \(y\) is the variable, and again, an exponent of 1 is implied.
By breaking down terms into these components, you can more easily manage and simplify algebraic expressions. Recognizing each part helps in determining the like terms and applying further simplification steps.
Basic Algebraic Principles
Algebra relies on foundational principles that guide us in simplification, solving, and understanding expressions and equations. Here are some basic principles every student must know:
  • Commutative Property: This principle states that the order of addition or multiplication does not affect the outcome. For example, \(a + b = b + a\) or \(ab = ba\).
  • Associative Property: This allows numbers to be grouped differently without changing their sum or product. As in \((a + b) + c = a + (b + c)\).
  • Distributive Property: This property is crucial for simplifying expressions. It states \(a(b + c) = ab + ac\), distributing the multiplication over the addition.
  • Identifying and Combining Like Terms: As covered in the previous section, recognizing like terms is key to simplifying expressions, as only like terms can be added or subtracted directly.
Returning to the expression \(9x + 3y\), it mainly involves understanding the commutative nature of addition but does not require further simplification due to the lack of like terms. Mastering these foundational principles equips you to handle a wide variety of algebraic problems with confidence, ensuring you apply the correct method and procedures in any given scenario.

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