Problem 105 - Beginning and Intermediate Algebra

Question

If \(x=-5\) and \(y=-3,\) evaluate each expression. $$ 2 x^{2}-y^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 41.

Step 1: Substitute Values into the Expression

First, substitute the given values of \(x\) and \(y\) into the expression. The expression is \(2x^2 - y^2\). So, replace \(x\) with \(-5\) and \(y\) with \(-3\), giving us: \[ 2(-5)^2 - (-3)^2 \].

Step 2: Calculate the Squares

Next, calculate the squares of \(-5\) and \(-3\). \(-5\) squared is \((-5)\times(-5) = 25\) and \(-3\) squared is \((-3)\times(-3) = 9\). Substitute these values into the expression: \[ 2(25) - 9 \].

Step 3: Multiply and Simplify

Now, multiply \(2\) by \(25\), which equals \(50\). Substitute back into the expression to simplify: \[ 50 - 9 \].

Step 4: Final Evaluation

Finally, perform the subtraction. Subtract \(9\) from \(50\) to get the final result: \[ 41 \].

Key Concepts

Evaluation of Expressions

Evaluating algebraic expressions involves finding the value of an expression by substituting given numbers for each variable. This process is fundamental in algebra, providing a practical way to work with formulas and equations.
To evaluate an expression, follow these steps:

• Identify the expression you are working with.
• Recognize the values assigned to each variable in the expression.
• Substitute these values into the expression, replacing each variable with its corresponding number.
• Perform the arithmetic operations in the expression to arrive at a numerical answer.

By carefully substituting and correctly following the order of operations—parentheses, exponents, multiplication and division, and addition and subtraction (often remembered as PEMDAS)—you ensure the accuracy of the evaluated expression.

Substitution Method

The substitution method is a key technique in mathematics, especially for solving equations or simplifying expressions. It involves replacing variables with given numerical values. In the context of evaluating expressions, like in the exercise, substitution is straightforward.
When using the substitution method:

• Clearly write down the expression you need to evaluate, identifying each variable.
• Replace every instance of the variable in the expression with the value you've been provided.
• After substituting, proceed with solving or simplifying the expression using basic arithmetic operations.

For example, in the given expression, when you substitute \(x = -5\) and \(y = -3\), you'll transform it from an abstract formula into one involving numerical calculations. Substitution not only simplifies the process but also aids in visualizing the outcome with concrete numbers.

Exponents and Powers

Understanding exponents and powers is crucial when working with algebraic expressions, as they frequently appear in both simple and complex expressions. An exponent indicates how many times a number, known as the base, is multiplied by itself.
For instance, \(x^2\) means \(x\) multiplied by itself. If you substitute \(x = -5\), this becomes \((-5)\times(-5)\), which equals \(25\).
Key things to remember about exponents:

• If the base is negative and the exponent is an even number, the result will be positive.
• If the base is negative and the exponent is an odd number, the result will be negative.
• Always perform the operation related to the exponent before moving on to other operations, following the order of operations.

Working with exponents requires careful calculation and attention to detail to ensure that calculations are done correctly. In the exercise, knowing how to calculate the square of a negative number helps in the correct evaluation of the expression.