Solving Linear Equations Step-by-Step Solution For X

In the realm of mathematics, solving linear equations is a fundamental skill. Linear equations, characterized by variables raised to the power of one, are ubiquitous in various fields, from basic algebra to complex engineering applications. Mastering the techniques to solve these equations is crucial for anyone seeking to excel in mathematics and related disciplines.

In this article, we will embark on a comprehensive journey through the world of linear equations, focusing on a specific example to illustrate the step-by-step process of finding the solution. We will delve into the core concepts, unravel the intricacies of equation manipulation, and equip you with the tools to confidently tackle a wide range of linear equations.

Understanding Linear Equations

Before we dive into the solution process, let's first establish a clear understanding of what linear equations are and their key properties. A linear equation is an algebraic equation in which the highest power of the variable is one. It can be represented in the general form:

ax + b = c

where:

• x is the variable we aim to solve for.
• a, b, and c are constants, which are known numerical values.

The goal of solving a linear equation is to isolate the variable x on one side of the equation, thereby determining its value. This involves performing algebraic operations on both sides of the equation while maintaining equality. The fundamental principle is that any operation performed on one side must also be performed on the other side to preserve the balance of the equation.

Deconstructing the Equation: A Step-by-Step Approach

Now, let's turn our attention to the specific equation presented:

10x - 4.5 + 3x = 12x - 1.1

This equation is a linear equation in one variable, x. Our mission is to find the value of x that satisfies this equation. To achieve this, we will follow a systematic approach, breaking down the problem into manageable steps.

1. Combining Like Terms: Simplifying the Equation

The first step in solving this equation is to simplify both sides by combining like terms. Like terms are terms that contain the same variable raised to the same power. In our equation, we have two terms with x on the left side: 10x and 3x. We can combine these terms by adding their coefficients:

10x + 3x = 13x

Substituting this back into the equation, we get:

13x - 4.5 = 12x - 1.1

2. Isolating the Variable Term: Bringing x to One Side

Next, we want to isolate the variable term, 13x, on one side of the equation. To do this, we can subtract 12x from both sides of the equation. Remember, whatever operation we perform on one side, we must perform on the other to maintain balance:

13x - 4.5 - 12x = 12x - 1.1 - 12x

This simplifies to:

x - 4.5 = -1.1

3. Isolating the Variable: Unveiling the Value of x

Now, we have x almost isolated. To completely isolate x, we need to eliminate the constant term, -4.5, from the left side. We can achieve this by adding 4.5 to both sides of the equation:

x - 4.5 + 4.5 = -1.1 + 4.5

This simplifies to:

x = 3.4

4. The Solution: x = 3.4

Therefore, the solution to the equation 10x - 4.5 + 3x = 12x - 1.1 is x = 3.4. This means that when we substitute 3.4 for x in the original equation, both sides of the equation will be equal.

Verifying the Solution: Ensuring Accuracy

To ensure the accuracy of our solution, it's always a good practice to verify it by substituting the value of x back into the original equation. Let's do that:

10(3.4) - 4.5 + 3(3.4) = 12(3.4) - 1.1

Simplifying both sides:

34 - 4.5 + 10.2 = 40.8 - 1.1

39.7 = 39.7

Since both sides of the equation are equal, our solution x = 3.4 is verified and correct.

Mastering Linear Equations: Practice Makes Perfect

Solving linear equations is a fundamental skill in mathematics, and like any skill, it requires practice. The more you practice, the more comfortable and confident you will become in solving various types of linear equations. Here are some tips to help you master this skill:

• Understand the basic principles: Make sure you have a solid grasp of the core concepts of linear equations, such as combining like terms, isolating the variable, and maintaining equality.
• Practice regularly: Set aside time each day or week to practice solving linear equations. The more you practice, the better you will become.
• Start with simple equations: Begin with simple equations and gradually work your way up to more complex ones. This will help you build your confidence and understanding.
• Check your solutions: Always verify your solutions by substituting them back into the original equation. This will help you identify any errors and ensure accuracy.
• Seek help when needed: If you encounter difficulties, don't hesitate to seek help from teachers, tutors, or online resources. There are many resources available to support your learning.

Applications of Linear Equations: Real-World Relevance

Linear equations are not just abstract mathematical concepts; they have numerous applications in real-world scenarios. From calculating distances and speeds to modeling financial growth and analyzing data, linear equations play a vital role in various fields.

Here are a few examples of how linear equations are used in real-world applications:

• Physics: Linear equations are used to describe the motion of objects, such as calculating the distance traveled by a car at a constant speed.
• Engineering: Engineers use linear equations to design structures, analyze circuits, and model systems.
• Economics: Linear equations are used to model supply and demand, calculate costs and profits, and analyze economic trends.
• Finance: Financial analysts use linear equations to calculate interest rates, loan payments, and investment returns.
• Computer science: Linear equations are used in computer graphics, data analysis, and machine learning.

By mastering the art of solving linear equations, you are not only enhancing your mathematical skills but also equipping yourself with a valuable tool for solving real-world problems.

Conclusion: Empowering Your Mathematical Journey

In this comprehensive guide, we have explored the world of linear equations, focusing on a step-by-step approach to solving them. We have dissected the equation 10x - 4.5 + 3x = 12x - 1.1, demonstrating the process of combining like terms, isolating the variable term, and ultimately unveiling the solution x = 3.4.

Remember, the key to mastering linear equations lies in understanding the core concepts, practicing regularly, and seeking help when needed. With dedication and perseverance, you can conquer linear equations and unlock a world of mathematical possibilities.

As you continue your mathematical journey, embrace the challenges, celebrate your successes, and never stop exploring the fascinating world of equations and their applications. The ability to solve linear equations is a valuable asset that will serve you well in various aspects of your academic and professional life.

Keep practicing, keep learning, and keep growing your mathematical prowess. The world of mathematics is vast and rewarding, and linear equations are just the beginning of an exciting adventure.

Keywords: Solving Linear Equations, Step-by-Step Solution, Isolating the Variable, Combining Like Terms, Verifying the Solution, Real-World Applications