Question

Determine whether the monomials are like terms. \(6 x Y\) and \(5 x y\)

Short answer

Explain your reasoning. Answer: No, the monomials \(6 x Y\) and \(5 x y\) are not like terms because, although they share the variable "x" with the same degree (1), the variables "Y" and "y" have different cases and are considered separate variables.

Step-by-step solution

Identify the variables and their degrees

In the first monomial, \(6 x Y\), the variables are \(x\) and \(Y\). In the second monomial, \(5 x y\), the variables are \(x\) and \(y\). Observe that in both monomials, the variable "x" has the same degree (1), but the variable "Y" in the first monomial and "y" in the second monomial have different cases.

Compare the variables and their degrees

We can see that the variable "x" is common in both monomials and has the same degree (1). However, the cases of the variables "Y" and "y" are different (one is uppercase, and the other is lowercase). This difference implies that they are considered separate variables and not the same.

Determine if the monomials are like terms

Since the variables "Y" and "y" are considered as different variables, the monomials \(6 x Y\) and \(5 x y\) are not like terms.

Key concepts

Understanding Monomials

Monomials are expressions that consist of a single term. They can include numbers, variables, and the product of numbers and variables. When you think of monomials, imagine them as the building blocks of more complex algebraic expressions.

There are some key characteristics of monomials:

• They do not have addition or subtraction signs connecting them.
• Examples of monomials include terms like \(7x\), \(-3a^2\), or simply \(4\).
• Each monomial can have coefficients (like the 6 in \(6xY\)), and may have variables.
  

Distinguishing between simple and compound expressions is crucial when learning algebra. When comparing monomials, it's essential to focus on their core components, as it determines if they can be considered like terms.

Exploring Variables

Variables are symbols used to represent numbers in mathematical expressions. They are typically denoted by letters such as \(x\), \(y\), or \(z\). Variables provide the flexibility to generalize and solve a wide array of problems.

Here are some important points about variables:

• Variables can take on different values; they are placeholders for numbers we do not know yet.
• They are case sensitive, meaning \(x\) is different from \(X\), just like \(y\) is different from \(Y\).
• In the monomials \(6xY\) and \(5xy\), notice one has variable \(Y\) while the other has \(y\), which fundamentally changes the nature of these expressions.
  

Understanding how variables work is essential for identifying when terms are indeed comparable or adjustable within equations.

The Degree of a Variable

In algebra, the degree of a variable in a term is an indicator of its power. For example, in the monomial \(x^2\), the degree of the variable \(x\) is 2.

Consider the following about the degree of variables:

• The degree helps determine the nature and size of the terms. In \(6x\), the degree of \(x\) is 1.
• In expressions with more than one variable, like \(xy^3\), the degree is assessed for each variable separately. Here, \(x\) has a degree of 1, and \(y\) has a degree of 3.
• The overall degree of a monomial can be calculated by adding up the degrees of all variables involved.
  

Recognizing these variations helps in comparing monomials and understanding the completeness of any given term.

Comparison of Variables

When comparing variables, it is crucial to understand their distinct qualities. Even variables that might look similar can behave differently based on both their degree and case.

To compare variables effectively:

• Check if the variables are identical in terms of both letter and case. As seen in the monomials \(6xY\) and \(5xy\), the change from an uppercase Y to a lowercase y means they cannot be considered like terms.
• Ensure the degree of the variables matches if you're evaluating them for similarity.
  

By ensuring a detailed comparison of each variable's characteristics, you can determine whether the terms align, thus establishing their status as like terms or not.