Problem 41 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 18 x^{2} y-14 x^{2} y-20 x^{2} y $$

Step-by-Step Solution

Answer

Question: Simplify the following expression: $$ 18x^2y - 14x^2y - 20x^2y $$ Answer: The simplified expression is: $$ -16x^2y $$

Step 1: Identify like terms

We have to look for terms that have the same variables raised to the same powers. In this case, we can see that all three terms have the common variables \(x^2y\).

Step 2: Combine like terms

Now, we can combine these like terms by adding or subtracting their coefficients (the numbers in front of the variables). The expression we want to simplify is: $$ 18x^2y - 14x^2y - 20x^2y $$

Step 3: Perform arithmetic operations

We can now perform the arithmetic operations on the coefficients of the common variable \(x^2y\). $$ (18 - 14 - 20)x^2y $$ Calculating the result: $$ (-16)x^2y $$

Step 4: Write the final simplified expression

The simplified expression after combining the like terms is: $$ -16x^2y $$

Key Concepts

Like Terms

In algebra, identifying "like terms" is essential when simplifying expressions. Like terms are terms that contain the same variables raised to the same powers. For instance, in the expression given in the exercise—\(18x^2y - 14x^2y - 20x^2y\)—each term is a like term because they all include the variable component \(x^2y\). It doesn't matter what coefficients (the numerical part of the term) are in front, as long as the variable portions are identical. Recognizing like terms is the first step in many algebra problems because it allows you to simplify the expression. This simplification makes it easier to understand and work with an equation. Always remember, the power of the variable must match exactly, not just the variable itself. Identifying like terms is crucial for the successful manipulation and resolution of algebraic expressions.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra that involves condensing expressions into their simplest form. When simplifying, one combines like terms, a key process highlighted in the exercise. The main objective is to make the expression as straightforward as possible.To simplify an expression like \(18x^2y - 14x^2y - 20x^2y\), you start by grouping the like terms. These terms already share the same variables and exponents, which means you are able to directly combine them by adding or subtracting their coefficients.

• Identify all the like terms in the expression, just as seen in the step-by-step solution.
• Combine them by performing arithmetic operations on the coefficients while keeping the variable part unchanged.

Applying these steps helps to streamline the expression and is a skill frequently used in solving more complex equations.

Arithmetic Operations

Arithmetic operations are basic calculations including addition, subtraction, multiplication, and division. In algebra, they are often used to simplify expressions by working with the coefficients of like terms. In the exercise, after identifying the like terms with the common variable \(x^2y\), we focus on the numerical coefficients of these terms: 18, -14, and -20. The process in this context is:

• Add the coefficients if the terms have plus signs between them.
• Subtract if negative signs are present, as in \(18 - 14 - 20\).

Once these operations are performed, we get a single coefficient. In our example, the arithmetic operation results in \(-16\), leading to the final combined term \(-16x^2y\). This is an essential step in simplifying expressions, as it reduces multiple terms to a single term, making the algebraic expression succinct and easier to work with.