91Ó°ÊÓ

Variable substitution is a fundamental concept in algebra where we replace variables in an expression with specific numerical values. This makes it easier to evaluate complex expressions, allowing us to compute a concrete number rather than working with abstract symbols.
In our exercise, we have been provided with values for two variables: \(x = -2\) and \(y = 3\). When asked to evaluate the expression \(-2x^{2}y^{2}\), the first step is to substitute these values into the expression. Here's what you'll get: replacing every instance of \(x\) with \(-2\), and every \(y\) with \(3\), the expression transforms to \(-2(-2)^{2}(3)^{2}\).
This step simplifies the expression into a form that's ready for further mathematical operations. The substitution step is crucial for transitioning from the world of symbols to actual numbers, paving the way for effective problem-solving.

The order of operations is a set of rules that dictate the sequence in which mathematical operations need to be carried out.
The acronym PEMDAS can help you remember the correct order:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the expression \(-2(-2)^{2}(3)^{2}\), after substituting the variables, we follow these steps:

• First, resolve the exponents: \((-2)^{2} = 4\) and \((3)^{2} = 9\)
• Next, multiply these values in the expression from left to right: \(-2 \times 4 \times 9\)
This systematic approach ensures accuracy in calculations and prevents errors. By adhering to PEMDAS, we avoid the common mistake of performing operations out of order, which can lead to incorrect results.

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. It's the unifying thread of many math concepts and helps form the foundation for equations and functions used in numerous scientific disciplines.
In algebra, expressions involve variables, numbers, and operators. For instance, the given expression, \(-2x^{2}y^{2}\), includes variables \(x\) and \(y\), coefficients like -2, and the use of exponentiation.
Key elements of algebra include:

• Variables: Symbols that represent numbers, allowing for general mathematical modeling.
• Coefficients: Numbers that multiply a variable (like -2 in our expression).
• Expressions: Combinations of variables, numbers, and operators (like multiplication, division).

Understanding algebra is essential for solving equations and modeling real-world situations. With a strong grasp of algebraic principles, one can extend to more complex topics such as functions, calculus, and beyond.