Solving 8x + 14(x - 5) = 260 A Step-by-Step Guide

Jul 9, 2025 by ADMIN 50 views

Solving the Linear Equation 8x + 14(x - 5) = 260: A Comprehensive Guide

Introduction

In the realm of mathematics, solving linear equations is a fundamental skill. Linear equations, characterized by a single variable raised to the power of one, form the backbone of numerous mathematical and real-world applications. This article delves into the step-by-step solution of the linear equation 8x + 14(x - 5) = 260, providing a comprehensive guide for students, educators, and anyone seeking to enhance their problem-solving abilities. By meticulously dissecting each stage of the solution, we aim to foster a deeper understanding of the underlying principles and techniques involved in solving linear equations.

Understanding Linear Equations

Before embarking on the solution, it's crucial to grasp the essence of linear equations. A linear equation is an algebraic equation in which each term is either a constant or a variable multiplied by a constant. The variable, typically represented by 'x', appears only to the first power. The goal in solving a linear equation is to isolate the variable on one side of the equation, thereby determining its value. This involves employing a series of algebraic operations, such as addition, subtraction, multiplication, and division, while maintaining the equality of both sides of the equation.

The Equation: 8x + 14(x - 5) = 260

The equation at hand, 8x + 14(x - 5) = 260, presents a classic example of a linear equation. It involves the variable 'x', constants, and parentheses, which necessitate the application of the distributive property. Our mission is to systematically unravel the equation, step by step, until we arrive at the solution for 'x'.

Step 1: Applying the Distributive Property

The first step in tackling the equation is to eliminate the parentheses. This is achieved by applying the distributive property, which states that a(b + c) = ab + ac. In our equation, the term 14(x - 5) requires distribution. Multiplying 14 by both 'x' and '-5' yields 14x - 70. The equation now transforms to:

8x + 14x - 70 = 260

The distributive property is a cornerstone of algebraic manipulation. It allows us to simplify expressions by removing parentheses, thereby paving the way for subsequent steps in the solution.

Step 2: Combining Like Terms

The next step involves simplifying the equation by combining like terms. Like terms are terms that contain the same variable raised to the same power. In our equation, 8x and 14x are like terms. Combining them yields 22x. The equation now becomes:

22x - 70 = 260

Combining like terms streamlines the equation, reducing its complexity and bringing us closer to isolating the variable 'x'.

Step 3: Isolating the Variable Term

Our objective is to isolate the term containing 'x' on one side of the equation. To achieve this, we need to eliminate the constant term, -70, from the left side. We accomplish this by adding 70 to both sides of the equation. This maintains the equality of the equation while effectively canceling out the -70 on the left side:

22x - 70 + 70 = 260 + 70

22x = 330

By adding 70 to both sides, we have successfully isolated the term containing 'x', bringing us closer to the final solution.

Step 4: Solving for x

The final step involves solving for 'x' by isolating it completely. The variable 'x' is currently multiplied by 22. To isolate 'x', we need to divide both sides of the equation by 22:

22x / 22 = 330 / 22

x = 15

Thus, we have arrived at the solution: x = 15. This value of 'x' satisfies the original equation, making it the solution.

Verification

To ensure the accuracy of our solution, we can substitute x = 15 back into the original equation and verify that both sides are equal:

8(15) + 14(15 - 5) = 260

120 + 14(10) = 260

120 + 140 = 260

260 = 260

The left side of the equation equals the right side, confirming that our solution, x = 15, is indeed correct.

Conclusion

In this comprehensive guide, we have meticulously dissected the solution of the linear equation 8x + 14(x - 5) = 260. By applying the distributive property, combining like terms, and systematically isolating the variable, we arrived at the solution x = 15. The verification step further solidified the accuracy of our result.

This step-by-step approach not only provides the solution but also imparts valuable problem-solving skills applicable to a wide range of mathematical scenarios. Mastering the techniques involved in solving linear equations is an essential stepping stone for success in more advanced mathematical concepts.

Whether you are a student grappling with algebra or an educator seeking to enhance your teaching methods, this guide serves as a valuable resource. By diligently practicing these techniques, you can confidently tackle linear equations and unlock a deeper understanding of the mathematical world.

Linear Equations: Linear equations are algebraic expressions where the highest power of the variable is one. These equations can be represented graphically as a straight line and are fundamental in algebra. Solving linear equations involves finding the value of the variable that makes the equation true.
Distributive Property: The distributive property is a key concept in algebra that allows you to multiply a single term by two or more terms inside a set of parentheses. It states that a(b + c) = ab + ac. This property is essential for simplifying expressions and solving equations.
Combining Like Terms: Combining like terms is a simplification technique used in algebra to group terms that have the same variable raised to the same power. For example, 3x + 2x can be combined to 5x. This process helps to make equations and expressions easier to work with.
Isolating the Variable: Isolating the variable is the goal when solving equations. It involves performing operations on both sides of the equation to get the variable alone on one side. This typically involves using inverse operations such as addition, subtraction, multiplication, and division.
Verification: Verification is the process of checking your solution to an equation by substituting it back into the original equation. If the equation holds true, then your solution is correct. This step is crucial to ensure accuracy in problem-solving.
Algebraic Manipulation: Algebraic manipulation refers to the techniques used to rearrange and simplify algebraic expressions and equations. It includes applying properties such as the distributive property, combining like terms, and using inverse operations to solve for variables.
Problem-Solving Skills: Problem-solving skills are the abilities to understand a problem, devise a plan, carry out the plan, and review the solution. These skills are crucial in mathematics and in many other areas of life. Practicing with equations helps to develop and refine these skills.
Mathematical Concepts: Mathematical concepts are the underlying principles and ideas in mathematics. They form the basis for more complex topics and are essential for understanding and applying mathematical knowledge. A strong grasp of fundamental concepts like linear equations is key to success in mathematics.
Educational Resource: An educational resource is any tool or material that supports teaching and learning. This can include textbooks, online tutorials, videos, and practice problems. High-quality educational resources are essential for effective learning in mathematics.
Step-by-Step Approach: A step-by-step approach involves breaking down a problem into smaller, manageable steps. This method is particularly useful in mathematics as it helps to clarify the process and reduce errors. Each step builds upon the previous one, leading to the final solution.

Additional Tips for Solving Linear Equations

Organize Your Work

Maintaining a neat and organized workspace is crucial for solving linear equations effectively. When tackling complex equations, it's easy to make mistakes if your work is disorganized. Use a clean sheet of paper, write each step clearly, and align equal signs vertically. This will help you keep track of your progress and reduce the likelihood of errors.

Check for Common Mistakes

One common mistake in solving linear equations is neglecting to apply operations to both sides of the equation. Remember, to maintain equality, any operation performed on one side must also be performed on the other side. Another common error is mishandling negative signs. Pay close attention to the signs of each term and ensure you are applying them correctly throughout the solution process.

Practice Regularly

Like any skill, solving linear equations requires practice. The more you practice, the more comfortable and confident you will become. Work through a variety of problems, starting with simpler equations and gradually progressing to more complex ones. Consistent practice will help you develop a deeper understanding of the underlying concepts and techniques.

Use Online Resources

Numerous online resources are available to help you practice and improve your skills in solving linear equations. Websites like Khan Academy, Mathway, and Symbolab offer tutorials, practice problems, and step-by-step solutions. These resources can be invaluable tools for reinforcing your understanding and identifying areas where you may need additional support.

Seek Help When Needed

If you are struggling to solve linear equations, don't hesitate to seek help. Talk to your teacher, a tutor, or a classmate. Explaining the problem to someone else can often clarify your thinking and help you identify the source of your difficulty. There are also many online forums and communities where you can ask questions and receive assistance from experienced math enthusiasts.

Apply Linear Equations to Real-World Problems

One of the best ways to solidify your understanding of linear equations is to apply them to real-world problems. Look for opportunities to use linear equations in everyday situations, such as calculating costs, determining distances, or managing budgets. This will help you see the practical relevance of the concepts you are learning and make them more meaningful.

Review Fundamental Concepts

Sometimes, difficulties in solving linear equations can stem from gaps in your understanding of fundamental concepts. Take the time to review basic algebraic principles, such as the order of operations, the distributive property, and combining like terms. A strong foundation in these concepts will make it easier to tackle more complex problems.

Use Different Methods to Solve the Same Problem

There are often multiple ways to solve a linear equation. Try using different methods to solve the same problem. This will not only help you deepen your understanding but also allow you to develop a flexible problem-solving approach. If you can consistently arrive at the correct solution using different methods, you can be confident in your understanding.

Reflect on Your Mistakes

Everyone makes mistakes, but it's how you learn from them that matters. When you make a mistake in solving a linear equation, take the time to analyze what went wrong. Identify the specific step where you made the error and try to understand why you made it. This will help you avoid making the same mistake in the future and strengthen your problem-solving skills.

Make it a Habit

Solving linear equations should become a habit. Integrate it into your daily or weekly routine. The more you engage with these types of problems, the more proficient you will become. This consistent engagement will build your confidence and make you a more effective problem solver in mathematics.

Conclusion

Solving the linear equation 8x + 14(x - 5) = 260 is a valuable exercise in algebra. By following a step-by-step approach, we can break down complex problems into manageable tasks. The distributive property, combining like terms, and isolating the variable are essential techniques that empower us to find solutions. Regular practice and a solid understanding of these concepts will help you excel in algebra and beyond.