Solving Linear Equations: A Comprehensive Guide to Finding the Value of X

Introduction

Linear equations are a fundamental part of algebra, with many practical applications in various fields. Understanding how to solve these equations is crucial for anyone looking to advance their mathematical skills. In this article, we will focus on the common and straightforward process of isolating the variable, specifically demonstrating this technique with an example equation: if 9x - 6 -3x 30, what is x? This guide is tailored to ensure that readers can grasp the concept and apply it effectively.

Step-by-Step Guide to Solving the Equation 9x - 6 -3x 30

Let's break down the steps involved in solving this equation, making it easily understandable and trustworthy for a wide audience.

Step 1: Isolating X

Our goal is to isolate the variable x. To do this, we need to eliminate all other terms on one side of the equation so that we can find the value of x. The first step is to gather all terms involving x on one side of the equation and all other terms on the other side.

Equation Transformation

Start with the given equation:

9x - 6 -3x 30

Move all terms involving x to the left side of the equation, and all other terms to the right side.

9x 3x 30 6

Simplify the left side:

12x 36

Step 2: Dividing to Find the Value of X

Now that we have simplified the equation to 12x 36, the next step is to divide both sides of the equation by 12 to solve for x. This step ensures that the value of x is isolated.

Dividing Both Sides by 12

12x / 12 36 / 12

x 3

Explanation of the Solution

The value of x that satisfies the equation 9x - 6 -3x 30 is 3. This means if you substitute x 3 back into the original equation, it will hold true. Let’s verify:

Left side: 9(3) - 6 27 - 6 21

Right side: -3(3) 30 -9 30 21

Since both sides are equal, our solution is correct.

Applications of Linear Equations

Linear equations have a wide range of practical applications. Here are a few examples:

Example 1: Budgeting

Suppose you have a monthly budget of $360, and you spend $240 on rent and $30 on groceries. If you want to save the rest, you can set up the equation:

x - 240 - 30 0

Solving for x (total earnings) gives you x 300 360 660. This shows you need to earn $660 to cover your expenses and save the rest.

Example 2: Distance and Time

If a car travels at a constant speed of 60 miles per hour, you can use linear equations to find out how long it takes to travel a certain distance. For example, if the distance is 180 miles, the equation would be:

Speed Distance / Time

60 180 / Time

Time 180 / 60 3 hours.

Best Practices for Solving Linear Equations

1. Verify Your Solution

Always substitute your solution back into the original equation to ensure it is correct.

2. Use a Calculator for Complex Equations

For more complex equations, using a calculator can help ensure accuracy.

3. Practice Regularly

The more you practice, the better you will become at recognizing patterns and solving equations quickly.

Conclusion

Mastering the process of isolating x in linear equations not only enhances your mathematical skills but also provides valuable tools for real-world problem-solving. Understanding these techniques can prove invaluable in various academic and professional settings. Whether you are a student, a professional, or simply interested in improving your mathematical proficiency, the steps outlined in this guide can serve as a reliable resource for solving linear equations.