Problem 44 - Algebra for College Students

Question

Evaluate the algebraic expressions in Problems 35-57 for the given values of the variables. \(x^{2} y^{3}-2 x y+x^{2} y^{2}, \quad x=-1\) and \(y=-3\)

Step-by-Step Solution

Answer

The value of the expression is -24.

Step 1: Substitute the Given Values

First, substitute \(x = -1\) and \(y = -3\) into the expression \(x^2 y^3 - 2xy + x^2 y^2\). This gives us:\((-1)^2 (-3)^3 - 2(-1)(-3) + (-1)^2 (-3)^2\).

Step 2: Calculate Each Term

Evaluate each term separately:1. \((-1)^2 (-3)^3 = 1(-27) = -27\).2. \(-2(-1)(-3) = -6\).3. \((-1)^2 (-3)^2 = 1(9) = 9\).

Step 3: Combine the Results

Combine the results from Step 2:\(-27 - 6 + 9\).

Step 4: Simplify the Expression

Simplify the combined expression:\(-27 - 6 + 9 = -33 + 9 = -24\).

Key Concepts

Substitution Method

The substitution method is a widely used technique in algebra that involves replacing variables in an equation with given numerical values. This method is incredibly helpful when you need to evaluate an expression at specific values of the variables. Let's look at how it works in our given problem.Consider the expression \(x^2 y^3 - 2xy + x^2 y^2\), and we are asked to evaluate it for \(x = -1\) and \(y = -3\). Using the substitution method, you simply replace every occurrence of \(x\) in the expression with \(-1\), and every \(y\) with \(-3\). This transforms the expression into a purely numerical calculation: \((-1)^2 (-3)^3 - 2(-1)(-3) + (-1)^2 (-3)^2\).The key steps in the substitution method include:

• Identifying the variables in the expression.
• Correctly substituting the given values for these variables.
• Becoming comfortable with moving back and forth between variable expressions and their numerical counterparts.
• Makes calculation of expression values straightforward, especially when trying out multiple possible values for variables.

By practicing this approach, you'll quickly be able to handle increasingly complex algebraic problems with confidence.

Polynomials

Polynomials are fundamental algebraic expressions that consist of variables raised to whole number powers and combined using addition, subtraction, and multiplication. They are a central part of algebra and show up in many different mathematical contexts.A polynomial can be something as simple as \(x - 3\), or as complex as our example: \(x^2 y^3 - 2xy + x^2 y^2\). Here, we have terms that involve both multiplication and powers of the variables \(x\) and \(y\). Each term of a polynomial has coefficients and variables of different degrees:

• The term \(x^2 y^3\) indicates a product of \(x\) squared and \(y\) cubed.
• The term \(-2xy\) is a linear product of \(x\) and \(y\).
• Finally, \(x^2 y^2\) involves squaring both \(x\) and \(y\).

Understanding polynomials involves recognizing patterns in these terms and using operations to simplify or evaluate them. The ability to manipulate polynomials is important because it forms the backbone of many concepts in algebra and calculus.

Simplification of Expressions

Simplification of algebraic expressions is the process of making an expression easier to work with. This is done by combining like terms, performing basic operations, and reducing the expression to its simplest form without changing its value. Simplifying expressions is a crucial skill in algebra as it makes further calculations and evaluations more manageable.In our exercise, after performing the substitution, we get the numerical expression \((-27) - 6 + 9\). To simplify, you should:

• Perform any arithmetic operations or combine terms that are similar.
• In our example, this involves first subtracting \(27\) from \(-6\), getting \(-33\), and then adding \(9\), leading to \(-24\).
• Always check your work for potential errors, especially signs or overlooked operations.

The simplification process often involves a step-by-step method of combining and calculating, which reduces complex expressions into a single, easily understood number or term. Mastering this technique allows you to take on more complicated problems with ease, breaking them down into bite-sized steps.