Problem 43 - Intermediate Algebra

Question

Evaluate the algebraic expressions for the given values of the variables. $$ 3 x y-x^{2} y^{2}+2 y^{2}, \quad x=5 \text { and } y=-1 $$

Step-by-Step Solution

Answer

The value of the expression is -38.

Step 1: Substituting Variables

First, substitute the given values of the variables into the algebraic expression. The expression is \(3xy-x^2y^2+2y^2\). Substitute \(x=5\) and \(y=-1\) into this expression, so it becomes \(3(5)(-1) - (5)^2(-1)^2 + 2(-1)^2\).

Step 2: Calculating Each Term

Now calculate each term of the expression using the substituted values. The first term is \(3(5)(-1) = -15\). The second term is \(-(5)^2(-1)^2 = -(25)(1) = -25\). The third term is \(2(-1)^2 = 2(1) = 2\).

Step 3: Adding the Results

Finally, combine all the calculated terms to find the value of the expression. Add \(-15\), \(-25\), and \(2\) together: \(-15 - 25 + 2 = -38\).

Key Concepts

Substituting Variables

When evaluating algebraic expressions, the first step is to substitute the given values for each variable in the expression. This means replacing every instance of the variable with the given number. For example, in the expression \(3xy - x^2y^2 + 2y^2\), we substitute \(x = 5\) and \(y = -1\).Here's how you do it:

• Find each occurrence of \(x\) and replace it with 5.
• Find each occurrence of \(y\) and replace it with -1.

The expression becomes easier to manage, hence offering a clearer path to simplify and evaluate further. The substituted expression for our example is \(3(5)(-1) - (5)^2(-1)^2 + 2(-1)^2\). By executing this substitution correctly, the problem already starts to seem less daunting.

Calculating Terms

After successfully substituting the values for the variables, the next step is to calculate each term of the expression. This involves handling the arithmetic operations like multiplication and exponentiation. Each term must be calculated separately to avoid errors.Follow these steps to calculate terms:

• For the term \(3(5)(-1)\), multiply the numbers in sequence to get \(-15\).
• For the term \(-(5)^2(-1)^2\), first calculate the exponents: \( (5)^2 = 25 \) and \((-1)^2 = 1\). Then multiply these results as \(-(25)(1) = -25\).
• For the term \(2(-1)^2\), calculate the exponent \((-1)^2 = 1\) and multiply by 2 to get 2.

By ensuring each term is calculated correctly, the expression becomes manageable as you move towards simplifying it.

Expression Simplification

The final step in evaluating an algebraic expression is to simplify it by combining the calculated terms. Simplification often involves adding or subtracting these terms to find a single numerical result.For our given example, once each term \(-15\), \(-25\), and \(2\) is calculated, you simply add them together:

• Add \(-15\) and \(-25\): the result is \(-40\).
• Add \(-40\) and \(2\) to get the final result of \(-38\).

This simplification process is a crucial step, as it provides the final answer by reducing the problem to the simplest form. By following these steps, any algebraic expression involving substitutions can be evaluated accurately and efficiently.