Problem 42 - Algebra 2

Question

Simplify. $$ \left(4 x^{2}-3 y^{2}+5 x y\right)-\left(8 x y+3 y^{2}\right) $$

Step-by-Step Solution

Answer

The simplified expression is \(4x^2 - 6y^2 - 3xy\).

Step 1: Distribute the Negative Sign

The problem is to simplify the expression \( (4x^2 - 3y^2 + 5xy) - (8xy + 3y^2) \). We start by distributing the negative sign to the terms inside the second parenthesis. This gives us \( 4x^2 - 3y^2 + 5xy - 8xy - 3y^2 \).

Step 2: Combine Like Terms

Now, we combine like terms. Start with the \( y^2 \) terms: \(-3y^2 - 3y^2\) which simplifies to \(-6y^2\). Next, combine the \( xy \) terms: \(5xy - 8xy\) is \(-3xy\). The \( x^2 \) term \(4x^2\) remains unchanged. Thus, the expression simplified is \(4x^2 - 6y^2 - 3xy\).

Key Concepts

Distributive Property

The distributive property is an essential algebraic concept used to simplify expressions. It allows us to remove parentheses by distributing a factor across terms inside the parentheses. In the expression \( (4x^2 - 3y^2 + 5xy) - (8xy + 3y^2) \), the negative sign outside the second set of parentheses is distributed to each term inside, effectively changing the sign of each term. Here's how it works:

• The negative sign multiplies each term inside \(- (8xy + 3y^2)\).
• This results in \(-8xy - 3y^2\).

Combining these, the expression becomes:\( 4x^2 - 3y^2 + 5xy - 8xy - 3y^2 \).Understanding the distributive property helps in algebraic manipulations, ensuring expressions are simplified accurately.

Like Terms

Like terms are terms in an algebraic expression that have identical variable parts raised to the same power. They can be combined by adding or subtracting their coefficients. This is key to simplifying expressions.In the expression \( 4x^2 - 3y^2 + 5xy - 8xy - 3y^2 \), like terms include:

• The \( y^2 \) terms: \(-3y^2\) and \(-3y^2\) which combine to \(-6y^2\).
• The \( xy \) terms: \( 5xy \) and \(-8xy\) which combine to \(-3xy\).
• The \( x^2 \) term \(4x^2\) which stands alone.

Understanding how to combine like terms simplifies complex algebraic expressions and is crucial for mathematical problem-solving.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (such as +, -, *, and /). In algebraic expressions, terms are separated by addition or subtraction.The original expression\( (4x^2 - 3y^2 + 5xy) - (8xy + 3y^2) \)is an example of a situation where simplifying is needed to remove unnecessary complexity. Simplification involves:

• Using the distributive property to eliminate parentheses, as explained previously.
• Identifying and combining like terms to achieve a simpler form.

The result of simplifying the given expression is a more straightforward version,\( 4x^2 - 6y^2 - 3xy \), that makes it easier to understand and use in further mathematical calculations. Understanding these components is vital for mastering algebra.