Problem 41 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(2 x^{2}-4 x y-3 y^{2}, \quad x=1\) and \(y=-1\)

Step-by-Step Solution

Answer

The value of the expression is 3.

Step 1: Substitute the Values

Start by substituting the given values, \(x = 1\) and \(y = -1\), into the algebraic expression \(2x^2 - 4xy - 3y^2\). This process will allow us to transform the expression using known values for the variables.

Step 2: Calculate Each Term

Now, calculate each part of the expression separately:- Compute \(2x^2\): \(2(1)^2 = 2\).- Compute \(-4xy\): \(-4(1)(-1) = 4\).- Compute \(-3y^2\): \(-3(-1)^2 = -3\).

Step 3: Sum the Results

After evaluating each term separately, combine them to form the final expression: \(2 + 4 - 3\).

Step 4: Simplify the Expression

Add the results from the previous step: \(2 + 4 - 3 = 3\).

Key Concepts

Understanding the Substitution Method

The substitution method is a fundamental tool in algebra that helps simplify expressions by replacing variables with known values. When dealing with an expression like \(2x^2 - 4xy - 3y^2\), and given specific values for \(x\) and \(y\), the substitution method allows you to plug these values directly into the expression.

• Identify the variables in the expression – here, they are \(x\) and \(y\).
• Replace each occurrence of \(x\) and \(y\) with their given values. For this problem, \(x = 1\) and \(y = -1\).

The main advantage of using substitution is that it transforms an abstract expression into a numerical problem, making it easier to manage and solve. It's like painting by numbers in algebraic terms — each letter is replaced by a specific number.

Variable Evaluation Simplified

Variable evaluation is the process of computing the result of an expression after substituting the values of the variables. Once we have replaced the variables using the substitution method, the next step is to evaluate the expression by calculating each term one by one.
To break it down:

• Calculate the \(2x^2\) term: Substitute, then compute \(2(1)^2 = 2\).
• Compute \(-4xy\): Replace the variables first, then solve \(-4(1)(-1) = 4\).
• Finally, find \(-3y^2\): This becomes \(-3(-1)^2 = -3\) after substitution and calculation.

Evaluation results in a series of numerical values that are easier to compute. This step helps demystify the expression by taking away the variables, leaving only numbers which are much simpler to add or subtract.

Simplifying Expressions for Clarity

Once you have evaluated all the terms separately, the last step is to simplify the expression. Simplification helps make the results clear and concise.
Begin by adding the evaluated results:

• Start with \(2\) from the \(2x^2\) term.
• Add \(4\) that resulted from \(-4xy\).
• Finally, subtract \(3\) from the \(-3y^2\) evaluation.

By performing these operations sequentially, \(2 + 4 - 3\), you simplify to \(3\). Simplifying expressions not only concludes the problem but also checks your work. If mistakes were made in calculations, the results often don’t align, indicating a need to review the previous steps. It's the final polish that brings clarity and confirmation to your mathematical work.