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The substitution method is a critical skill in algebra that makes evaluating expressions straightforward. In this method, you replace variables with their given values. This is very helpful when you have expressions with multiple variables.
Just imagine you're swapping parts of a puzzle! For example, when you have an expression \( \frac{x-y}{3z} \), and you're told that \( x = -2 \), \( y = 7 \), and \( z = -3 \), you can simply plug these numbers into the expression. This means that wherever you see \( x \), you replace it with -2, \( y \) with 7, and \( z \) with -3.
Hence, the substitution makes the expression \( \frac{-2 - 7}{3(-3)} \). It helps in simplifying complex algebraic problems into basic arithmetic operations.

In algebraic expressions, understanding fractions is critical. A fraction consists of two parts: the numerator and the denominator.
Think of a fraction like a cake. The numerator is the piece of the cake you want, while the denominator tells you how many pieces the cake is divided into. In our example \( \frac{x-y}{3z} \), after substitution, the numerator is \(-2 - 7\) and the denominator is \(3(-3)\).

• Numerator: The top part of the fraction, representing the value we're interested in.
• Denominator: The bottom part of the fraction, representing the value by which we divide the numerator.

Understanding both parts enables us to perform arithmetic operations correctly. By identifying and understanding these parts, we can simplify and solve the expression more effectively.

Arithmetic operations are the basic computations we perform on numbers, like addition, subtraction, multiplication, and division.
In this context, after substituting the values, the next step is to deal with the arithmetic operations in \( \frac{-2 - 7}{3(-3)} \). Here's how you can proceed:

• First, perform the subtraction in the numerator: \(-2 - 7\) which equals \(-9\).
• Next, handle the multiplication in the denominator: \(3 \times (-3)\) which is also \(-9\).

Now, you have \( \frac{-9}{-9} \). These simple arithmetic operations lay the foundation for solving and simplifying expressions.

Simplifying expressions makes it easier to understand and use their values in other mathematical operations.
Here, simplification involves reducing \( \frac{-9}{-9} \) into its most basic form. Since both the numerator and the denominator are \(-9\), division results in \(1\). This is because dividing a number by itself gives you one. It’s like having a whole pie and dividing it fairly among one person - you get the whole pie!
Simplifying expressions involves:

• Breaking down complex problems.
• Making calculations more manageable.
• Arriving at a single, clear answer.

By simplifying, we convert the problem into a more understandable form, aiding in comprehending algebraic expressions better.