Problem 74 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ x y(3 x y+2 x-5 y)-2 x^{2} y^{2}-5 x^{2} y+4 x y^{2} $$

Step-by-Step Solution

Answer

Question: Simplify the expression: \(x y (3 x y + 2 x - 5 y) - 2 x^{2} y^{2} - 5 x^{2} y + 4 x y^{2}\) Answer: \(x^{2} y^{2} - 3 x^{2} y - x y^{2}\)

Step 1: Apply distributive property

Distribute the \(x y\) term inside the parenthesis: $$ x y (3 x y + 2 x - 5 y) = 3 x^{2} y^{2} + 2 x^{2} y - 5 x y^{2} $$

Step 2: Combine the terms outside the parenthesis with the result from Step 1

Now, add or subtract the terms with the same variables and powers: $$ 3 x^{2} y^{2} + 2 x^{2} y - 5 x y^{2} - 2 x^{2} y^{2} - 5 x^{2} y + 4 x y^{2} $$

Step 3: Simplify and combine like terms

Combine the terms in the expression that have the same variables and powers: $$ (3 x^{2} y^{2} - 2 x^{2} y^{2}) + (2 x^{2} y - 5 x^{2} y) + (-5 x y^{2} + 4 x y^{2}) $$ Perform the operations within each parenthesis: $$ x^{2} y^{2} - 3 x^{2} y - x y^{2} $$ The simplified expression is: $$ x^{2} y^{2} - 3 x^{2} y - x y^{2} $$

Key Concepts

Distributive Property in Algebra

The distributive property is an essential principle in algebra that helps in simplifying expressions by breaking them down into smaller parts. This property states that for any numbers or variables, \( a(b + c) = ab + ac \). In the context of the given exercise, we use this property to simplify the expression \( x y(3 x y + 2 x - 5 y) \).

The idea is to distribute the term on the outside of the parentheses, \(xy\), to each term inside the parentheses. Think of it as sharing or spreading out the multiplication over each term within the parentheses. Doing this, our expression becomes:

• \(3x^2y^2\) (from multiplying \(xy \) with \(3xy \)),
• \(2x^2y\) (from multiplying \(xy \) with \(2x \)),
• \(-5xy^2\) (from multiplying \(xy \) with \(-5y \)).

By distributing like this, we can handle each term separately, making the task of simplifying the expression much easier.

Combining Like Terms

Once you have distributed terms across an expression using the distributive property, the next step is to combine like terms. Like terms are terms in an expression that have identical variable parts raised to the same power. For example, \(3x^2y^2\) and \(2x^2y^2\) are like terms, as are \(2x^2y\) and \(-5x^2y\).

To approach this, you start by identifying terms that share the same variables and exponents. In the expression:

• \(3 x^{2} y^{2}\) and \(-2 x^{2} y^{2}\) are grouped together,
• \(2 x^{2} y\) and \(-5 x^{2} y\) are grouped together,
• \(-5 x y^{2}\) and \(4 x y^{2}\) are grouped together.

After sorting them, calculate the result of each group. By reverting like terms to a simpler form, the expression is easily manageable and less prone to error.

Simplification of Expressions

Simplifying an expression is often the last step in solving an algebra problem. This process involves combining like terms as well as ensuring that the expression is written in the simplest form. The aim is to make expressions as short as possible.

In our exercise, after distributing and combining like terms, you simplify what remains:

• Combine \(3x^2y^2\) and \(-2x^2y^2\) to get \(x^2y^2\).
• Combine \(2x^2y\) and \(-5x^2y\) to get \(-3x^2y\).
• Finally, combine \(-5xy^2\) and \(4xy^2\) to get \(-xy^2\).

The combination of these operations results in the simplified expression \(x^2 y^2 - 3 x^2 y - x y^2\).

By practicing these steps, you'll master the simplification process and will be able to approach even more complex algebraic expressions with confidence.