Problem 42 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The evaluated expression is 26.

Step 1: Substitute the values

Replace the variables in the expression with the given values. Here, substitute \(x = 3\) and \(y = -2\) into:\[4x^2 + xy - y^2\] This gives:\[4(3)^2 + (3)(-2) - (-2)^2\]

Step 2: Simplify each term

Calculate each term separately:1. \(4(3)^2 = 4 \times 9 = 36\)2. \(3(-2) = -6\)3. \(-(-2)^2 = -4\) (since \((-2)^2 = 4\))

Step 3: Combine the terms

Add the results of the simplified terms together:\[36 + (-6) - 4 = 36 - 6 - 4\]

Step 4: Perform the final calculation

Calculate the result from the combined terms:\[36 - 6 = 30\]\[30 - 4 = 26\]

Key Concepts

Substitution in Algebra

Substitution in algebra is like replacing variables with actual numbers. This makes it easier to evaluate expressions and find numerical answers to problems. In our exercise, we have an algebraic expression: \[4x^2 + xy - y^2\]To evaluate it, we substitute specific values for the variables given in the problem. Here, we replace \(x\) with 3 and \(y\) with -2. By substituting these values into the expression, it transforms into a series of arithmetic calculations that are more straightforward to solve. Key points to remember:

• Identify the variable values given in the problem.
• Replace each instance of the variable in the expression with its respective value.
• Be mindful of negative signs and squared terms, as they can affect the outcome drastically.

Simplifying Expressions

Simplifying expressions involves transforming a complex expression into a simpler form without changing its value. After substitution, each part of the expression can be evaluated independently to simplify your work. This might involve using basic arithmetic operations like multiplication, addition, or subtraction. For the given expression:\[4(3)^2 + (3)(-2) - (-2)^2\]we first simplify each term:

• \(4(3)^2\) becomes \(4 \times 9 = 36\)
• \(3(-2)\) results in \(-6\)
• \(-(-2)^2\) equals \(-4\), since \((-2)^2\) yields 4 but must be negated

Only after simplifying each part should you move on to combining them, ensuring fewer chances for mistakes.

Combining Like Terms

Combining like terms is the final step where you gather all simplified terms into a single numerical result. Once you've simplified the expression, you'll work your way through by performing the arithmetic operations in the correct order. This process ensures clarity and correctness in the final evaluation.Here’s how it works for the problem:The terms from simplifying the expression were 36, -6, and -4. Combine them as follows:

• Begin with \(36\) as your base number.
• Add \(-6\) to \(36\), resulting in \(30\).
• Then subtract \(4\) from \(30\) to arrive at our final answer of \(26\).

This step-by-step combination not only helps in getting the correct answer but also ensures that each part of the expression contributes correctly to the final result.