Problem 75 - Elementary Algebra Within Reach

Question

In Exercises \(75-86\), simplify the expression. $$ x^{2}-2 x y+4+x y $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( x^{2} -xy + 4 \).

Step 1: Recognize Like Terms

In the given expression, \( x^{2} - 2xy + 4 + xy \), we can recognize that \( -2xy \) and \( xy \) are like terms that can be combined.

Step 2: Combine Like Terms

When we combine \( -2xy \) and \( xy \), we get \( -2xy + xy = -xy \).

Step 3: Rewrite the Simplified Expression

Replacing \( -2xy + xy \) with \( -xy \) in the original expression gives us the simplified form: \( x^{2} -xy + 4 \).

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, one of the most crucial steps is to combine like terms. Like terms are terms in an expression that have the exact same variables raised to the same power, even though they may have different coefficients.
For example, in the expression \( x^2 - 2xy + 4 + xy \), \( -2xy \) and \( xy \) are like terms. They both have the same variables \( x \) and \( y \) raised to the first power, thus making it possible to combine them. You can think of like terms as similar ingredients in a recipe; just as you would combine two cups of flour into one quantity, you can combine these terms into a single term.
How do we combine them? By simply adding or subtracting their coefficients, the numerical part of the terms. For \( -2xy \) and \( xy \) (which has a hidden coefficient of 1), the combination results in \( -xy \), because \( -2 + 1 = -1 \). It's important to pay special attention to the signs (positive or negative) of the coefficients when combining like terms.

Algebraic Simplification

The process of algebraic simplification involves rewriting expressions in a more concise and comprehensible manner without changing their value. The ultimate goal is to make the expression as straightforward as possible, so evaluating or further manipulating it becomes easier.
To simplify an algebraic expression, you should follow a systematic approach:

• Identify and combine like terms, as discussed earlier.
• Apply the distributive property if applicable, to remove parentheses.
• Reduce any complex fractions.
• Combine any constants (numbers without variables).

The expression from our exercise \( x^2 - 2xy + 4 + xy \) becomes \( x^2 - xy + 4 \) after simplification. Notice that this new form is easier to understand and use in solving equations or inequalities. Consistent, careful simplification is a fundamental skill in algebra that ensures higher accuracy and proficiency in solving more complex problems.

Elementary Algebra

Elementary algebra is the branch of mathematics that introduces the concepts of variables and constants and the rules for manipulating them. With its language and operations, algebra allows us to formulate equations and inequalities and to solve a wide range of problems.
Central to understanding elementary algebra is recognizing the role of variables, which are symbols (often letters) that represent numbers, and constants, which are fixed numerical values. In our example, \( x \) and \( y \) are variables while \( 4 \) is a constant.
This foundation underlies all algebraic manipulation, including simplifying expressions like the one in our exercise. As you grow more familiar with combining like terms and simplifying expressions, you'll find that elementary algebra becomes not just a set of rules to follow, but a powerful tool for solving problems and understanding mathematical relationships.