Problem 42 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ 4 x^{2}+x y-y^{2}, \quad x=3 \text { and } y=-2 $$

Step-by-Step Solution

Answer

The value of the expression is 26.

Step 1: Substitute the given values into the expression

Take the given algebraic expression \(4x^2 + xy - y^2\) and substitute \(x = 3\) and \(y = -2\) into it. This gives us: \(4(3)^2 + (3)(-2) - (-2)^2\).

Step 2: Calculate the squares

Calculate \(3^2\) and \((-2)^2\). We have \(3^2 = 9\) and \((-2)^2 = 4\). So the expression becomes: \(4(9) + (3)(-2) - 4\).

Step 3: Multiply the coefficients

Now multiply the coefficients: \(4 \times 9 = 36\) and \(3 \times (-2) = -6\). The expression now is: \(36 - 6 - 4\).

Step 4: Simplify the expression

Perform the addition and subtraction: \(36 - 6 = 30\) and then \(30 - 4 = 26\). So, the expression simplifies to \(26\).

Key Concepts

Substitution Method

The substitution method is an essential technique for dealing with algebraic expressions, especially when evaluating them for specific values of variables. The main goal is to replace variables with their given numerical values. This transforms an algebraic expression into a numerical one.
To use this method:

• First, identify the variables in the expression and their corresponding values, as given in the problem.
• Replace each variable in the expression with its value. This requires attention to signs and coefficients to maintain accuracy.
• Ensure that every instance of the variables is substituted to simplify further calculations.

In our example, we substitute the values \(x = 3\) and \(y = -2\) into the expression \(4x^2 + xy - y^2\), resulting in a purely numerical expression: \(4(3)^2 + (3)(-2) - (-2)^2\). This step is crucial as it sets the stage for mathematical calculations without the complexities of variables.

Evaluating Expressions

Evaluating expressions is the process of computing the eventual numerical value of an algebraic expression after substituting the given values of its variables.

After substitution, calculate the results of any operations within the expression, following the correct order of operations. This involves performing calculations step-by-step:

• Calculate any exponents first as they dictate the power to which a number is raised. In our example, compute \(3^2\) and \((-2)^2\) because these are the square operations required.
• Proceed with multiplication or division. Multiply the results or values accordingly as seen in the expression.
• Finally, carry out any addition or subtraction as the last step. Ensure operations are performed sequentially from left to right.

Evaluating \(4x^2 + xy - y^2\) gives us \(4(9) + (3)(-2) - 4\). Further simplifying results in \(36 - 6 - 4\). This highlights the importance of methodical computation to reach the correct answer, which is \(26\) in this case.

Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division, combined with exponentiation, are the building blocks of evaluating any mathematical expression. Each operation has its place and importance, especially when evaluated in a specific order, known as the order of operations (PEMDAS/BODMAS).

Here's a breakdown of the operations:

• **Exponents:** Represents repeated multiplication of a number with itself. Important to handle early in expressions to avoid errors.
• **Multiplication and Division:** These operations are performed after exponents and from left to right. They simplify parts of the expression, affecting its core value.
• **Addition and Subtraction:** These operations come last, again performed from left to right. They adjust the overall tally of the expression's numerical result.

For our example, after substituting and simplifying, operations were carried out as:

• Compute exponents: \(3^2 = 9\) and \((-2)^2 = 4\).
• Perform multiplication: \(4 \times 9 = 36\), \(3 \times (-2) = -6\).
• Finalize with subtraction and addition: \(36 - 6 = 30\) and then \(30 - 4 = 26\).

By adhering to these operations, one can navigate through expressions effectively, ensuring accuracy at every step.