Problem 53 - Algebra A Combined Function

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5 .\) $$ 3 x-2 $$

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Understand the Expression

The expression given is \(3x - 2\), and it is required to evaluate this expression using the provided values for the variables: \(x = 1\).

Step 2: Substitute the Value for x

Substitute \(x = 1\) into the expression \(3x - 2\). This becomes \(3(1) - 2\).

Step 3: Perform the Multiplication

Calculate the multiplication \(3 \times 1 = 3\). The expression now simplifies to \(3 - 2\).

Step 4: Perform the Subtraction

Subtract 2 from 3: \(3 - 2 = 1\).

Step 5: Final Result

The expression \(3x - 2\) evaluates to 1 when \(x = 1\).

Key Concepts

Substitution Method

Substitution is a useful technique in algebra where we replace a given variable with a specific value to simplify and solve expressions. For example, in the expression \(3x - 2\), if we know the value of \(x\), like \(x=1\) in this case, we can directly substitute 1 wherever we see \(x\) in the expression.

Here's how you can use the substitution method efficiently:

• Identify the variables in your mathematical expression.
• Replace each variable with its given value. This requires placing the value inside parentheses to clarify operations, such as \(3(1)\) instead of \(31\).

This approach helps in transforming the expression into a more manageable arithmetic problem, allowing you to continue solving using basic operations.

Arithmetic Operations

Arithmetic operations are the basic calculations we perform with numbers: multiplication, division, addition, and subtraction. In our example, after substituting the value into the expression \(3x - 2\), we are left with a simple arithmetic problem: \(3(1) - 2\).

Let's break it down step-by-step:- **Multiplication:** First, calculate \(3 \times 1\) to get 3. This is derived from substituting and then multiplying the coefficient with the value of \(x\).- **Subtraction:** Next, we perform the subtraction. Subtract 2 from the result of the previous multiplication, which is 3 - 2. This operation simplifies the expression to 1.Understanding how to perform these operations correctly is crucial, as each step follows logically from the previous one and they are foundational for solving algebraic expressions.

Evaluating Expressions

Evaluating expressions means finding the value of an algebraic expression by carrying out the required arithmetic operations. The main idea is to simplify the expression as much as possible, step by step, using any given values for variables.

Here's a simple guide to evaluating:

• **Perform Substitution:** Begin by substituting values for all variables in the expression. This turns an algebraic expression into a numeric one.
• **Simplify Step-by-Step:** Perform each arithmetic operation according to the order of operations (PEMDAS/BODMAS) to avoid errors. Simplify the expression until it cannot be simplified further.

In our problem, we evaluated \(3x - 2\) by substituting, multiplying, and then subtracting, which gave us the final value of 1. By mastering these steps, we can approach more complex expressions with confidence.