91Ó°ÊÓ

In mathematical expressions, the substitution method is a powerful tool for simplification and evaluation. It's like a puzzle, where each piece must be correctly placed to solve the picture. When given specific values for variables, you replace each variable in an expression with its corresponding value. This process helps remove the abstract nature of algebra, anchoring variables to actual numbers.

For instance, in our expression, we are given the values:

• \( x = -2 \)
• \( y = 7 \)
• \( z = -3 \)

With these values, substitute as follows into the expression \( x^{2} - 2z^{2} + 4xy \):
\( (-2)^{2} - 2(-3)^{2} + 4(-2)(7) \).

This initial step lays the foundation for understanding and proceeding with further calculations. By substituting numbers into their respective places, you create a simpler form that is easier to work with.

Simplifying expressions is about breaking them down into manageable parts. The goal is to make expressions more straightforward by performing arithmetic operations in a particular order. This process, often governed by the rules of PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), helps prevent mistakes and ensures clarity.

Let's simplify the expression already set up by substitution:\((-2)^{2} - 2(-3)^{2} + 4(-2)(7)\).
Follow these steps:

• Start with Exponents: \( (-2)^{2} = 4 \) and \( (-3)^{2} = 9 \).
• Next, handle the Multiplication: \(2 \times 9 = 18 \) and \(4 \times (-2) \times 7 = -56 \).
• Finally, tackle the Subtraction and Addition: Simplifying to \(4 - 18 - 56\).

At this point, continue adding and subtracting to reach the final simplified value: \(-70\). Simplifying expressions boils down to organizing calculation steps and confirming each part is reduced as much as possible.

Evaluating expressions is the final step, where the simplified expression is reduced to a single numerical value. This process provides meaning and a concrete answer to abstract algebraic constructs. Evaluation involves executing arithmetic on the expression where all variables are already replaced and simplified.

After substitution and simplification, we have reached:

• \( 4 - 18 - 56 \)
• Compute each arithmetic operation: \( 4 - 18 = -14 \)
• Then compute: \(-14 - 56 = -70 \)

Thus, the expression \(x^{2} - 2z^{2} + 4xy\) evaluates to \(-70\) when \(x = -2\), \(y = 7\), and \(z = -3\). Evaluating expressions is ultimately about finding that one neat, final answer that sums up the entire mathematical journey. This final step verifies that all substituting and simplifying were done correctly, arriving at a precise value, which confirms our understanding of the relationship among numbers and operations in the expression.