Problem 25
Question
$$ 9 g $$$$ 5 x y+3 x $$
Step-by-Step Solution
Verified Answer
Answer: The simplest forms of the expressions are "9g" and "5xy + 3x".
1Step 1: Task 1: Simplify the expression "9g"
In this expression, there's only one term which is "9g". There are no like terms to combine, so the expression is already in its simplest form. The answer is 9g.
2Step 2: Task 2: Simplify the expression "5xy + 3x"
In this expression, we have two terms, "5xy" and "3x". Since they have different variable parts (xy and x), they are not like terms and cannot be combined. Therefore, the expression is already in its simplest form. The answer is 5xy + 3x.
Key Concepts
SimplificationLike and Unlike TermsVariables
Simplification
Simplifying an algebraic expression is one of the fundamental skills in mathematics. It refers to the process of making an expression as simple as possible by applying mathematical rules efficiently.
When simplifying, we aim to combine any like terms and reduce the expression to its most basic form. Sometimes, an expression is already simplified, meaning it cannot be made simpler.
For instance, in the expression \(9g\), there is only one term. Since there are no terms to combine with \(9g\), it's already in its simplest form.
In other cases, when dealing with more terms, simplification involves checking if they can be combined. It's essential for solving equations and helps in understanding relations and changes within an expression.
When simplifying, we aim to combine any like terms and reduce the expression to its most basic form. Sometimes, an expression is already simplified, meaning it cannot be made simpler.
For instance, in the expression \(9g\), there is only one term. Since there are no terms to combine with \(9g\), it's already in its simplest form.
In other cases, when dealing with more terms, simplification involves checking if they can be combined. It's essential for solving equations and helps in understanding relations and changes within an expression.
Like and Unlike Terms
In algebra, terms are the different parts of an expression that are separated by plus or minus signs. To effectively simplify expressions, understanding the difference between like and unlike terms is crucial.
**Like terms** are terms within an expression that have the same variable parts raised to the same power. This means you can combine them by adding or subtracting the coefficients. For example, \(3x\) and \(5x\) are like terms because they both contain the same variable \(x\).
In contrast, **unlike terms** have different variable parts or powers, and therefore, cannot be combined into a single term. For example, \(5xy\) and \(3x\) are unlike terms because they have different variable components (\(xy\) vs. \(x\)).
Recognizing like and unlike terms allows you to effectively simplify and manage algebraic expressions, especially when solving equations.
**Like terms** are terms within an expression that have the same variable parts raised to the same power. This means you can combine them by adding or subtracting the coefficients. For example, \(3x\) and \(5x\) are like terms because they both contain the same variable \(x\).
In contrast, **unlike terms** have different variable parts or powers, and therefore, cannot be combined into a single term. For example, \(5xy\) and \(3x\) are unlike terms because they have different variable components (\(xy\) vs. \(x\)).
Recognizing like and unlike terms allows you to effectively simplify and manage algebraic expressions, especially when solving equations.
Variables
In algebra, variables are symbols used to represent unknown values or values that can change. They are often represented by letters like \(x\), \(y\), or \(g\).
Variables allow us to write general formulas and expressions without specifying exact numbers, giving expressions the ability to represent real-world scenarios and adaptable situations.
For instance, in the expressions \(9g\) and \(5xy + 3x\), \(g\), \(x\), and \(y\) are variables that could represent any number.
Understanding how to manipulate variables and recognize their roles within expressions is key. This becomes especially important when you solve algebraic equations, as you often need to isolate these variables to find their values.
Variables allow us to write general formulas and expressions without specifying exact numbers, giving expressions the ability to represent real-world scenarios and adaptable situations.
For instance, in the expressions \(9g\) and \(5xy + 3x\), \(g\), \(x\), and \(y\) are variables that could represent any number.
Understanding how to manipulate variables and recognize their roles within expressions is key. This becomes especially important when you solve algebraic equations, as you often need to isolate these variables to find their values.
Other exercises in this chapter
Problem 25
For the following problems, find the products. $$ (4 x+2)^{2} $$
View solution Problem 25
For the following problems, simplify each of the algebraic expressions. $$ a+8 a+3 a $$
View solution Problem 25
Perform the following additions and subtractions. $$ 6 y^{2}+2 y-1+\left(5 y^{2}-18\right) $$
View solution Problem 25
Use numerical evaluation on the equations. Geometry (area of a triangle) \(A=\frac{1}{2} b h . \) Find \(A\) if \(b=25\) and \(h=10\).
View solution