Problem 52 - Elementary Algebra

Question

Simplify. $$ x 2+y 2-2 x y-x 2+5 x y-y_{2} $$

Step-by-Step Solution

Answer

The expression simplifies to \( 3xy \).

Step 1: Reorganize the Terms

The given expression is \( x^2 + y^2 - 2xy - x^2 + 5xy - y^2 \). Let's first reorganize the terms by grouping like terms together. Rewrite the expression: \( (x^2 - x^2) + (y^2 - y^2) + (-2xy + 5xy) \).

Step 2: Simplify Grouped Terms

Now, let's simplify each group of terms. First, simplify \( x^2 - x^2 \), which equals 0. Then, simplify \( y^2 - y^2 \), which also equals 0. Lastly, simplify \( -2xy + 5xy \), which results in \( 3xy \).

Step 3: Write the Simplified Expression

Combining all the simplified groups together, you are left with the expression \( 3xy \).

Key Concepts

Understanding Like Terms

A key concept in algebra is the idea of 'like terms.' These are terms in an expression that can be combined because they have the same variables raised to the same power. Recognizing and combining like terms is crucial in simplifying expressions effectively.

For example, let's look at the terms in our expression:

• In both terms \( x^2 \) and \( -x^2 \), the variable \( x \) is raised to the same power of 2.
• Similarly, \( y^2 \) and \( -y^2 \) are like terms because the \( y \) is squared in both.
• Notice \( -2xy \) and \( 5xy \) are also like terms because both have the variables x and y to the first power.

Recognizing these like terms allows us to simplify the expression by combining them, which brings us to the next concept.

Simplifying Expressions

Simplifying expressions involves combining like terms to create a simpler, more manageable expression. This process often involves both reordering the terms and using basic arithmetic.

In our expression, \( x^2 + y^2 - 2xy - x^2 + 5xy - y^2 \), we start by organizing like terms together:

• \( (x^2 - x^2) \) results in 0 because they cancel each other out.
• \( (y^2 - y^2) \) also results in 0 because they cancel out.
• Finally, \( (-2xy + 5xy) \) simplifies to \( 3xy \), as -2 plus 5 equals 3.

After completing these steps, the simplified form of the expression is \( 3xy \). Breaking down expressions into manageable steps can make complex algebra much clearer.

Exploring Polynomials

The expressions we are dealing with are examples of polynomials, which are mathematical expressions consisting of variables and coefficients. Polynomials can have different degrees and multiple terms.

The original expression, \( x^2 + y^2 - 2xy - x^2 + 5xy - y^2 \), is a polynomial with terms that have variables x and y raised to either the first or second power.

• The degree of a polynomial is determined by the term with the highest sum of exponents. In our case, the degree is 2 because of the terms \( x^2 \) and \( y^2 \).
• Polynomials can be simplified to reveal the simplest form, like in our example, where the final polynomial is \( 3xy \), showcasing the power of simplification.

Understanding polynomials and simplifying them presents a clear pathway to efficiently handling complex algebraic expressions.