Solving Linear Equations: Step-by-Step Guide with Real-life Applications

Linear equations are fundamental concepts in algebra, often encountered in various real-life applications. In this article, we will dive into the solution of a simple linear equation, 3x - 3 3, and explore the steps to solve such equations in detail.

Understanding and Solving the Equation: 3x - 3 3

Consider the equation: 3x - 3 3. The goal is to isolate the variable x on one side of the equation. Let's begin by adding 3 to both sides of the equation to eliminate the -3 on the left side:

Step-by-Step Solution

Step 1: Add 3 to Both Sides

Adding 3 to both sides of the equation.

Starting with the original equation:

3x - 3 3

Adding 3 to both sides:

3x - 3   3  3   3

Simplifies to:

3x 6

Step 2: Divide by the Coefficient of x

To isolate x, we need to divide both sides of the equation by the coefficient of x, which is 3:

3x/3  6/3

This simplifies to:

x 2

Verification

Let's verify the solution by substituting x 2 back into the original equation:

3(2) - 3 3

Which simplifies to:

6 - 3 3

Thus:

3 3

This confirms that our solution, x 2, is correct.

Why Follow the Rules?

When solving equations, it's crucial to apply the same operation to both sides of the equation to maintain equality. This ensures that the equation remains balanced and the solution is accurate. Always verify your solution by plugging it back into the original equation.

General Steps to Solve Linear Equations

Isolate the Variable

Move all terms with the variable to one side of the equation. Move all numbers to the other side of the equation. Combine like terms if necessary. Divide by the coefficient of the variable to isolate the variable.

Example: x - 3 3 x

Let's solve another linear equation: x - 3 3 x.

Bring x on one side: x - x - 3 3 x - x This simplifies to: -3 3 -3 - 3 0 -6 0

This is a contradiction, indicating that there is no solution to the equation.

Conclusion

Mastering the process of solving linear equations is crucial for students and professionals alike. Understanding the fundamental steps and rules ensures accuracy and reliability in solving such equations. Practice with real-life applications will further enhance your problem-solving skills.

Key Points to Remember:

Balance the Equation: Apply the same operation to both sides. Isolate the Variable: Move all terms with the variable to one side and all numbers to the other. Verify: Always check your solution by substituting it back into the original equation.