Problem 37 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ 4 x^{2}-y^{2}, x=2 \text { and } y=-2 $$

Step-by-Step Solution

Answer

The value of the expression is 12.

Step 1: Substitute Given Values into the Expression

Start by substituting the given values of the variables into the expression. The expression is \(4x^2 - y^2\). Here, \(x = 2\) and \(y = -2\). Substitute these values to obtain: \(4(2)^2 - (-2)^2\).

Step 2: Calculate the Individual Terms

Evaluate each term separately. First, solve \(4(2)^2\):\[(2)^2 = 4\] \[4(4) = 16\]Next, solve \((-2)^2\):\[(-2)^2 = 4\]

Step 3: Combine the Calculated Values

After calculating the individual terms, subtract the second term from the first:\[16 - 4 = 12\]

Key Concepts

Evaluating Expressions

Evaluating algebraic expressions involves replacing the variables in an expression with given numerical values and simplifying it to reach a final answer. This process helps in obtaining a specific value from an expression, depending on the variables' values provided.
In our example, we're dealing with the expression \(4x^2 - y^2\). Here, the variables are \(x\) and \(y\), for which specific values are given: \(x = 2\) and \(y = -2\). Evaluating expressions is crucial because:

• It transforms abstract numbers and operations into tangible results.
• It is essential for solving real-world problems in mathematics, engineering, and sciences.

To evaluate, substitute the known values into the equation and perform arithmetic operations.

Substitution in Algebra

Substitution is a fundamental concept in algebra that allows us to replace variables with their respective numerical values. This process simplifies complex expressions and equations, making them easier to solve or evaluate.
In the given problem, we substitute \(x = 2\) and \(y = -2\) into the expression \(4x^2 - y^2\). By substituting, the expression becomes \(4(2)^2 - (-2)^2\). This substitution step is important because:

• It simplifies calculations, leading to accurate results.
• It helps in visualizing how changes in variable values affect the expression.

This practical technique is widely used in mathematical problem-solving to quickly evaluate expressions under different scenarios.

Step-by-Step Solutions

A step-by-step approach is particularly useful for solving algebraic expressions, ensuring clarity and accuracy. By breaking down each part of the solution, you can identify and correct potential mistakes. Let's review the solution to our example:
1. **Substitution:** Begin by substituting the given values into \(4x^2 - y^2\). Replace \(x\) with \(2\) and \(y\) with \(-2\), transforming the expression to \(4(2)^2 - (-2)^2\).
2. **Individual Calculation:** Calculate each term separately. First, solve \(4(2)^2\): - \((2)^2 = 4\) - \(4 \times 4 = 16\)
Then, solve \((-2)^2\): - \((-2)^2 = 4\)
3. **Combine Values:** Subtract the second term from the first to get the final result: - \(16 - 4 = 12\)
This organized method ensures that each step follows logically from the previous one, fostering a deeper understanding of the problem-solving process.