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Substitution is a fundamental concept in mathematics where numbers or other expressions are inserted into variables. In the context of evaluating algebraic expressions, this means replacing variables—often single letters like \(x\) or \(y\)—with specific numbers.
Let's consider the expression \((xy + 1)^2\). Given the values \(x = 2\), \(y = 5\), and \(z = 3\), we start by substituting \(2\) for every \(x\) and \(5\) for every \(y\) in the expression.

• This results in: \((2 \cdot 5 + 1)^2\).

Substitution simplifies expressions and sets the stage for further calculations, allowing us to solve and understand how different parts of the expression relate. It's like swapping placeholders with real numbers to make the expression solvable.

The order of operations is a rule used to clarify which procedures should be performed first in a given mathematical expression. Without it, the same expression could yield different results.
To remember the correct order, the acronym PEMDAS is often used which stands for:

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

Applying this to our expression \((2 \cdot 5 + 1)^2\):

• First, handle the multiplication: \(2 \cdot 5 = 10\).
• Then, add \(1\) to \(10\) to get \(11\).
• Finally, square \(11\) to obtain \(121\).

This consistent method ensures everyone evaluates expressions the same way, just like following a recipe for identical results.

Evaluating algebraic expressions involves finding the numerical value of an expression by simplifying it through substitution and applying the order of operations. Given the expression \((xy + 1)^2\) and the values \(x = 2\) and \(y = 5\), we substitute and simplify step by step.
First, substitute the variables with their given numbers: \((2 \cdot 5 + 1)^2\).
Next, follow the order of operations to simplify inside the parentheses:

• Calculate the multiplication: \(2 \cdot 5 = 10\).
• Add \(1\): \(10 + 1 = 11\).

Finally, apply the exponent: Square \(11\) to get \(121\).
Evaluating expressions is a way to translate algebraic notation into actual numbers, using both substitution and a structured series of operations to arrive at a clear result.