Problem 97 - Elementary Algebra

Question

For Problems \(93-106\), evaluate each algebraic expression for the given values of the variables. $$ x^{2}-2 x y+y^{2} \quad \text { for } x=-\frac{1}{2} \text { and } y=2 $$

Step-by-Step Solution

Answer

The value of the expression is \( \frac{25}{4} \).

Step 1: Understand the Expression

We need to evaluate the algebraic expression \( x^2 - 2xy + y^2 \) by substituting the given values \( x = -\frac{1}{2} \) and \( y = 2 \).

Step 2: Substitute the Values

Substitute \( x = -\frac{1}{2} \) and \( y = 2 \) into the expression: \[ (-\frac{1}{2})^2 - 2 \cdot (-\frac{1}{2}) \cdot 2 + (2)^2 \]

Step 3: Calculate Each Term Separately

Compute each term separately:1. \( (-\frac{1}{2})^2 = \frac{1}{4} \)2. \(-2 \cdot (-\frac{1}{2}) \cdot 2 = 2 \)3. \( (2)^2 = 4 \)

Step 4: Substitute the Calculated Values Back

Substitute the computed values back into the expression:\[ \frac{1}{4} + 2 + 4 \]

Step 5: Simplify the Expression

Add the values together:\( \frac{1}{4} + 2 + 4 = \frac{1}{4} + \frac{8}{4} + \frac{16}{4} = \frac{25}{4} \)

Key Concepts

Substitution in Algebra

One of the most fundamental techniques in algebra is the substitution method. This involves replacing variables in an algebraic expression with their given numerical values to compute the result. In this particular exercise, we were tasked with substituting the variables \( x = -\frac{1}{2} \) and \( y = 2 \) into the expression \( x^2 - 2xy + y^2 \).

• Substitution simplifies the expression by converting it into a numerical equation.
• This helps in focusing purely on the arithmetic, bypassing any symbolic variables.

To carry out substitution effectively:- Identify the variables and their corresponding values.- Carefully replace each variable in the expression.- Ensure to include parentheses, which help in maintaining the order of operations.By applying this method, we moved from the abstract expression to practical numbers, setting the stage for simplification.

Understanding Exponents

Exponents in algebra represent repeated multiplication of a number by itself. In our expression, the terms \( x^2 \) and \( y^2 \) require us to understand this concept deeply.For instance, \( x^2 \) means \( x \cdot x \). When substituting \( x = -\frac{1}{2} \), we compute:- \((-\frac{1}{2})^2 = (-\frac{1}{2}) \cdot (-\frac{1}{2}) = \frac{1}{4}\).Similarly, \( y = 2 \) and \( y^2 = 2 \cdot 2 = 4 \).

• Positive numbers, when squared, remain positive.
• Negative numbers become positive, as a negative multiplied by a negative is positive.

Understanding exponents helps in simplifying complex expressions and accurately computing results after substitution, as both high powers or negative numbers can impact outcomes greatly.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of reducing them to their most concise form. After substitution and computing exponent values, the final step is bringing together all the terms.In our exercise, after calculating each term:- \( (-\frac{1}{2})^2 = \frac{1}{4} \)- \(-2 \cdot (-\frac{1}{2}) \cdot 2 = 2 \)- \( 2^2 = 4 \)These individual parts need to be added: - \( \frac{1}{4} + 2 + 4 \)Align fractions and whole numbers for addition:

• Convert 2 to \( \frac{8}{4} \).
• Convert 4 to \( \frac{16}{4} \).

Thus, the expression simplifies to \( \frac{1}{4} + \frac{8}{4} + \frac{16}{4} = \frac{25}{4} \).Simplification involves adding and aligning terms, ensuring calculations are easy and accurate, yielding the final solution of the algebraic problem.