Question

In each case, simplify the given expression, if possible. \( \quad 4 y+2 x-3 x y\)

Short answer

Question: Simplify the expression, if possible: \(4y + 2x - 3xy\). Answer: The simplified expression is: \(4y + 2x - 3xy\).

Step-by-step solution

Identify Similar Terms

The given expression is \(4y + 2x - 3xy\). Observe that there are no similar terms that can be combined because each term has distinct variables.

Factor if Possible

Given the expression \(4y+2x-3xy\), there is no common factor other than 1 in all the terms. So, factoring is not possible in this case.

Write the Simplified Expression

Since there are no similar terms to combine and the expression cannot be factored, the simplified expression is as it is. The simplified expression is: \(4y + 2x - 3xy\).

Key concepts

Simplification

Simplification in algebra involves rewriting an expression into its simplest form. In mathematics, the main goal is to make the expression easier to work with, without altering its value. In this exercise, we began with the expression

• \(4y + 2x - 3xy\)

To simplify, the first step is identifying similar terms that can be combined. Similar terms share the same variable with the same power, but in this case, the expression contains:

• One term with only \(y\)
• One term with only \(x\)
• One term with both \(x\) and \(y\)

Thus, there are no similar terms, and no combination is possible here. If it were possible, you would add or subtract the coefficients (the number in front of the variable) of similar terms. For example, like terms such as \(2x\) and \(5x\) would become \(7x\). In this exercise, because the terms differ in variable type, the simplified version is the same as the original:

• \(4y + 2x - 3xy\)

Factoring

Factoring is one of the key components of simplifying algebraic expressions, involving taking a common factor out of an expression to rewrite it in a different form. It’s akin to turning an expression inside out by looking for anything all terms share in common, like a common factor. Consider factors like numbers you multiply together to get another number.
For example, the factors of 6 are 2 and 3 because 2 * 3 = 6; these would be the common elements shared among terms if any were present. However, in this exercise with

• \(4y + 2x - 3xy\)

There is no number or variable that is common across all three terms. The only factor they have in common is 1, meaning they cannot be factored any further without adding complexity instead of reducing it.
Thus, while factoring can often lead to a more compact form, in this case, the expression remains as is because there's nothing common among all terms that can be pulled out.

Terms in Algebra

In algebra, understanding terms is crucial as they form the core components of expressions. A term can consist of numbers, variables, or the product of numbers and variables. In expressions like

• \(4y + 2x - 3xy\)

There are three separate terms:

• \(4y\): Consists of the constant 4 and variable \(y\)
• \(2x\): Consists of the constant 2 and variable \(x\)
• \(-3xy\): A product of constant -3 and variables \(x\) and \(y\)

It is important to observe how each term is composed. Coefficients (the number part like 4, 2, -3) determine the magnitude of each term, while the variables (\(x\), \(y\)) establish the term's identity. Terms can also contain just a single number, known as a constant, but our expression featured none.
The ability to identify and understand terms helps you evaluate if they can be grouped or factored. By knowing what each part of an expression represents, simplifying them becomes a simpler task.