91Ó°ÊÓ

Variable substitution is a fundamental process in algebra where we replace variables with their given numeric values. This allows us to simplify and evaluate algebraic expressions. When you have an expression such as \(3 + 2z\), and you know that \(z = 3\), you substitute 3 in place of \(z\) in the expression.
Consider these steps for variable substitution:

• Identify the variables in the expression.
• Substitute each variable with its given value.
• Simplify the expression if possible.

In our exercise, substituting \(z = 2\) transforms the expression from \(3 + 2z\) to \(3 + 2 \times 2\). This makes it easier to carry out further calculations.

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. These expressions are manipulated using rules of arithmetic, adjusting them until they are fully simplified or evaluated. Understanding the structure of algebraic expressions is crucial:

• Terms: These are the parts of the expression separated by '+' or '-' signs.
• Coefficients: Numbers that are multiplied by the variables.
• Constants: Numbers that stand alone without variables.

In our example, \(3 + 2z\), the expression consists of the constant \(3\) and the term \(2z\), where \(2\) is the coefficient of \(z\). Recognizing each piece allows for systematic evaluation through substitution and calculation.

A step-by-step calculation approach is invaluable for breaking down complex expressions into manageable parts. Here’s how to effectively tackle problems with this approach using our example expression \(3 + 2z\):
1. **Substitute Variables:** First, as discussed, replace \(z\) with \(2\), transforming the expression to \(3 + 2 \times 2\).2. **Calculate Individual Operations:** Next, handle multiplications or divisions first. So, calculate \(2 \times 2 = 4\).3. **Simplify Further:** Finally, perform any additions or subtractions remaining, resulting in \(3 + 4 = 7\).
By following this method, you ensure accuracy and clarity in solving algebraic expressions. Practicing this technique will also improve overall problem-solving skills in mathematics.