Problem 37 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 12.

Step 1: Substitute the Values

First, substitute the given values of the variables into the expression. The expression is \(4x^2 - y^2\). Substituting \(x = 2\) and \(y = -2\) into the expression, we have \(4(2)^2 - (-2)^2\).

Step 2: Evaluate the Squared Terms

Now, calculate the squares in the expression. For \(4(2)^2\), calculate \((2)^2\) first, which equals \(4\), and then multiply by \(4\) to get \(16\). For \((-2)^2\), the result is \(4\) since a negative number squared is positive.

Step 3: Simplify the Expression

Replace the squared terms in the expression with their respective values. So, now the expression becomes \(16 - 4\). Simplify this by subtracting: \(16 - 4 = 12\).

Key Concepts

Substitution in Algebra

Substitution is a technique used in algebra to make expressions easier to handle. It involves replacing variables with known values. This process transforms an expression involving variables into a more straightforward arithmetic computation.

To substitute:

• Identify the given values for each variable. In the example provided, we have \( x = 2 \) and \( y = -2 \).
• Replace each occurrence of the variable in the expression with its corresponding value. For our expression \(4x^2 - y^2\), this means substituting \( x \) with \( 2 \) and \( y \) with \(-2\).
• Write the new expression using these values: \(4(2)^2 - (-2)^2\).

Remember, every opportunity to substitute can simplify your work and pave the way for easier calculations. Always double-check that you've substituted all instances of each variable!

Squaring Numbers

Squaring a number means multiplying the number by itself. It is an essential operation in algebra that allows us to transform expressions and numbers into their squared equivalents.

Here is how to square numbers properly:

• For positive numbers like \(2\), squaring the number \(2\) means \((2)^2 = 2 \times 2 = 4\).
• When dealing with negative numbers, like \(-2\), squaring works the same way: \((-2)^2 = (-2) \times (-2) = 4\). Notice that the product is positive because multiplying two negative numbers yields a positive result.

Remember to calculate these squared terms separately to avoid errors. Squaring is straightforward but crucial for solving many algebraic expressions, so practicing this step helps ensure accuracy.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression more concise, ideally in its simplest form. This involves combining like terms and performing arithmetic operations.

Here's how to simplify expressions once substitution and squaring are done:

• First, place the calculated values back into your expression. From our example, after substitution and squaring, we have the expression \(16 - 4\).
• Now, perform the arithmetic operations. Start with any addition or subtraction involved. In this case, subtract \(4\) from \(16\) to get \(12\).

Having a simplified expression makes it easy to understand the core result of your calculations. Simplification is a frequent final step in solving equations and helps ensure clarity and correctness in your solution.