Problem 81 - Algebra A Combined Function

Question

Solve each equation. $$ 3 x+1=0 $$

Step-by-Step Solution

Answer

The solution is \( x = -\frac{1}{3} \).

Step 1: Isolate the variable term

The equation you have is: \( 3x + 1 = 0 \). The first step is to isolate the \( 3x \) term by subtracting 1 from both sides of the equation. This will give us: \( 3x = -1 \).

Step 2: Solve for the variable

Next, solve for \( x \) by dividing both sides of the equation by 3. From the equation \( 3x = -1 \), after dividing both sides by 3, you will get: \( x = -\frac{1}{3} \).

Key Concepts

Isolating Variables

When solving linear equations, isolating the variable is a crucial step. This means you want to get the term with the variable by itself on one side of the equation. Imagine you have the equation \( 3x + 1 = 0 \). The variable here is \( x \), and our goal is to find its value. To isolate \( 3x \), we need to remove the 1 from the left. We do this by subtracting 1 from both sides of the equation. This step gives us a simpler equation: \( 3x = -1 \).

In general, when isolating variables, remember to:

• Identify the term to isolate.
• Perform the same operation on both sides to maintain equality.
• Simplify the equation as much as possible.

This technique is a foundational skill in algebra, making it easier to solve equations accurately.

Equation Solving Steps

Solving a linear equation involves a series of systematic approaches. The purpose is to find the value of the variable that makes the equation true. Let's go through the process using our example, \( 3x + 1 = 0 \).

1. **Look at the equation:** Identify the variable and combine like terms if necessary.
2. **Isolate the variable term:** As previously discussed, use operations to get the variable term alone. Here, we subtracted 1 to get \( 3x = -1 \).
3. **Solve for the variable:** Once the variable term is isolated, proceed to solve for the variable. Divide both sides by 3: this transforms \( 3x = -1 \) into \( x = -\frac{1}{3} \).

These steps emphasize a logical sequence, ensuring clarity and accuracy in solving any linear equation.

Basic Algebra Concepts

Algebra is filled with basic concepts that ease into more complex topics. Linear equations and operations like addition, subtraction, multiplication, and division are at its core. Consider the equation \( 3x + 1 = 0 \). Each component represents a different aspect of basic algebra concepts.

- **Equations:** An equation states that two expressions are equal, and our goal is to discover the unknown variable that satisfies this balance.
- **Operations:** Basic operations allow us to manipulate equations. In our example, subtraction was used to isolate the term \( 3x \), and division was used to solve for \( x \).
- **Variables:** A variable is a symbol that represents an unknown number. It can take different values, and our task is to find that specific value that makes the equation true.

Understanding these foundational concepts equips you with the tools needed to solve a wide range of algebraic problems.