Solve The Equation 7x - 10 = 4x + 5. Show Each Step.

Introduction

In this detailed guide, we will explore the process of solving a linear equation. Linear equations, fundamental in algebra, involve finding the value of a variable that makes the equation true. Specifically, we will tackle the equation ${ 7x - 10 = 4x + 5 }$. This equation represents a classic example of a linear equation with one variable. Understanding how to solve such equations is crucial for various mathematical applications and problem-solving scenarios. The techniques we will use here are applicable to a wide range of similar problems, making this a valuable skill to acquire.

Solving linear equations requires a systematic approach. We aim to isolate the variable, in this case, x, on one side of the equation. This involves performing algebraic operations on both sides of the equation to maintain equality while simplifying the expression. These operations include addition, subtraction, multiplication, and division. By carefully applying these operations, we can gradually simplify the equation until we arrive at the solution.

The steps we will follow are designed to be clear and easy to understand. We will begin by outlining the initial equation, then proceed step-by-step to isolate x. Each step will be explained in detail, ensuring that you grasp the underlying logic. This method not only helps in solving this specific equation but also provides a solid foundation for tackling more complex algebraic problems in the future. Whether you are a student learning algebra for the first time or someone looking to refresh your skills, this guide will provide you with the tools and knowledge needed to confidently solve linear equations.

Step-by-Step Solution

1. State the Original Equation

The first step in solving any equation is to clearly state the original equation. This ensures that we have a reference point and helps prevent errors in subsequent steps. In our case, the original equation is:

${ 7x - 10 = 4x + 5 }$

Writing down the equation clearly helps to organize your thoughts and allows you to visualize the problem more effectively. This is especially important in more complex problems where multiple steps are involved. By starting with a clear statement of the equation, we set the stage for a systematic and accurate solution.

2. Isolate Terms with ${ x }$ on One Side

The next step involves isolating the terms containing the variable x on one side of the equation. To achieve this, we need to eliminate the x term from the right side of the equation. We can do this by subtracting ${ 4x }$ from both sides of the equation. This maintains the balance of the equation while moving the x terms to the left side:

${ 7x - 10 - 4x = 4x + 5 - 4x }$

Simplifying both sides, we get:

${ 3x - 10 = 5 }$

This step is crucial because it groups the variable terms together, making it easier to isolate x in the subsequent steps. By subtracting ${ 4x }$ from both sides, we have effectively consolidated the x terms on the left side, bringing us closer to the solution.

3. Isolate the Constant Terms on the Other Side

Now that we have the x terms on one side, we need to isolate the constant terms on the other side. In our equation, the constant term on the left side is -10. To eliminate this, we add 10 to both sides of the equation. This keeps the equation balanced and moves the constant term to the right side:

${ 3x - 10 + 10 = 5 + 10 }$

Simplifying both sides, we obtain:

${ 3x = 15 }$

This step is essential for further simplifying the equation and bringing us closer to isolating x. By adding 10 to both sides, we have successfully moved the constant term to the right side, leaving only the term with x on the left side.

4. Solve for ${ x }$

The final step is to solve for x. We have the equation ${ 3x = 15 }$. To isolate x, we need to divide both sides of the equation by the coefficient of x, which is 3:

${ \frac{3x}{3} = \frac{15}{3} }$

Simplifying both sides, we get:

${ x = 5 }$

Therefore, the solution to the equation ${ 7x - 10 = 4x + 5 }$ is ${ x = 5 }$. This is the value of x that makes the equation true. This final step completes the process of solving the linear equation, providing us with the answer we were seeking.

Verification of the Solution

To ensure the accuracy of our solution, it is always a good practice to verify the result. This involves substituting the value we found for x back into the original equation and checking if both sides of the equation are equal. This process helps to catch any errors that may have occurred during the solving process.

1. Substitute ${ x = 5 }$ into the Original Equation

Our original equation was:

${ 7x - 10 = 4x + 5 }$

Substitute ${ x = 5 }$ into the equation:

${ 7(5) - 10 = 4(5) + 5 }$

2. Simplify Both Sides

Now, we simplify both sides of the equation:

Left side:

${ 7(5) - 10 = 35 - 10 = 25 }$

Right side:

${ 4(5) + 5 = 20 + 5 = 25 }$

3. Compare the Results

We can see that both sides of the equation simplify to 25:

${ 25 = 25 }$

Since both sides are equal, our solution ${ x = 5 }$ is correct. This verification step confirms that we have accurately solved the equation. By substituting the value back into the original equation and confirming equality, we can be confident in our solution.

Conclusion

In summary, we have successfully solved the linear equation ${ 7x - 10 = 4x + 5 }$ by following a systematic approach. The steps we took included isolating the terms with x on one side, isolating the constant terms on the other side, and then solving for x. We found that the solution to the equation is ${ x = 5 }$. Additionally, we verified our solution by substituting ${ x = 5 }$ back into the original equation and confirming that both sides are equal.

Solving linear equations is a fundamental skill in algebra, and the methods we have used here are applicable to a wide range of similar problems. By understanding the principles of isolating variables and maintaining the balance of the equation, you can confidently tackle more complex algebraic challenges. The ability to solve linear equations is not only crucial for success in mathematics but also has practical applications in various fields, including science, engineering, and economics.

The key takeaways from this guide are the importance of a systematic approach, the necessity of performing the same operations on both sides of the equation, and the value of verifying the solution. By following these principles, you can improve your problem-solving skills and achieve greater accuracy in your mathematical work. Whether you are a student learning algebra or someone looking to refresh your skills, mastering the art of solving linear equations is a valuable asset.

This comprehensive guide has provided you with the knowledge and steps necessary to solve linear equations effectively. Remember to practice these techniques regularly to reinforce your understanding and build confidence in your ability to solve algebraic problems. With consistent effort and a clear understanding of the underlying principles, you can excel in mathematics and beyond.