Solving Linear Equations Step-by-Step A Comprehensive Guide

Introduction

In the realm of mathematics, solving linear equations is a fundamental skill. Linear equations, at their core, represent relationships between variables and constants, often depicting straight lines when graphed. Mastering the techniques to solve these equations is essential not only for academic success but also for various real-world applications. From calculating finances to determining scientific measurements, linear equations play a pivotal role in problem-solving. This comprehensive guide delves into the step-by-step process of solving the linear equation $-\frac{1}{2}(-10 x-10)-8=4(3 x-2)-9$, providing a clear and accessible approach for learners of all levels. We'll explore each operation, explaining the underlying principles and rationale behind every move. Furthermore, we'll emphasize the importance of checking the solution to ensure accuracy, a crucial step in the problem-solving process. By the end of this guide, you'll have a solid understanding of how to tackle similar equations and confidently navigate the world of linear algebra. This understanding will not only aid you in your mathematical journey but also equip you with valuable analytical skills applicable across various disciplines. The ability to manipulate and solve equations is a cornerstone of mathematical literacy, empowering you to interpret data, make predictions, and solve real-world problems with greater confidence. So, let's embark on this journey of mathematical exploration and unlock the secrets of solving linear equations.

Breaking Down the Equation: Step-by-Step Solution

To effectively solve the linear equation $-\frac{1}{2}(-10 x-10)-8=4(3 x-2)-9$, we will employ a systematic approach, carefully executing each step to arrive at the correct solution. This process involves simplifying both sides of the equation, isolating the variable, and verifying the result. Let's dissect the equation and unravel its solution, one step at a time.

1. Distribute Constants:

The initial step in solving this equation is to eliminate the parentheses by distributing the constants outside them. This involves multiplying the constant by each term within the parentheses. On the left side, we distribute $-\frac{1}{2}$ across $(-10x - 10)$. Remember that multiplying a negative number by a negative number results in a positive number. On the right side, we distribute 4 across $(3x - 2)$. Performing these multiplications carefully is crucial for maintaining the equation's balance and ensuring an accurate solution. Errors in distribution can cascade through subsequent steps, leading to an incorrect final answer. By meticulously applying the distributive property, we set the stage for further simplification and ultimately, the isolation of the variable. This foundational step is the cornerstone of solving algebraic equations and demonstrates the importance of precision in mathematical operations. Mastering the distributive property is a key skill in algebra and serves as a building block for more complex equation-solving techniques.

$-\frac{1}{2} * -10x = 5x$

$-\frac{1}{2} * -10 = 5$

$4 * 3x = 12x$

$4 * -2 = -8$

The equation now becomes:

$5x + 5 - 8 = 12x - 8 - 9$

2. Combine Like Terms:

After distributing the constants, the next logical step is to simplify both sides of the equation by combining like terms. Like terms are those that share the same variable and exponent, or are simply constants. On the left side, we can combine the constant terms $5$ and $-8$. On the right side, we combine $-8$ and $-9$. This simplification process makes the equation more manageable and easier to solve. By reducing the number of terms on each side, we reduce the complexity of the equation and bring it closer to a form where the variable can be easily isolated. The ability to identify and combine like terms is a fundamental algebraic skill that streamlines the equation-solving process. This step not only simplifies the equation but also reveals the underlying structure and relationships between the terms, making the subsequent steps more intuitive and efficient. It's a crucial step in maintaining clarity and preventing errors as we move towards the final solution.

$5 - 8 = -3$

$-8 - 9 = -17$

The equation is now simplified to:

$5x - 3 = 12x - 17$

3. Isolate the Variable Term:

Our goal is to isolate the variable, x, on one side of the equation. To achieve this, we need to move all terms containing x to one side and all constant terms to the other. A common strategy is to move the term with the smaller coefficient of x to the side with the larger coefficient to avoid dealing with negative coefficients unnecessarily. In this case, we can subtract $5x$ from both sides of the equation. This operation maintains the equation's balance and moves the x terms to the right side. Simultaneously, we can add 17 to both sides to move the constant terms to the left side. These two operations, performed in conjunction, effectively segregate the variable and constant terms, paving the way for the final isolation of x. This strategic manipulation of the equation is a hallmark of algebraic problem-solving, demonstrating the power of inverse operations in isolating unknowns. By carefully applying these techniques, we bring the equation closer to its solution, revealing the value of x in a clear and concise manner.

Subtract $5x$ from both sides:

$5x - 3 - 5x = 12x - 17 - 5x$

Which simplifies to:

$-3 = 7x - 17$

Add 17 to both sides:

$-3 + 17 = 7x - 17 + 17$

This further simplifies to:

$14 = 7x$

4. Solve for the Variable:

Now that we have isolated the variable term, $7x$, on one side of the equation, the final step in solving for x is to divide both sides of the equation by the coefficient of x, which is 7. This operation effectively isolates x and reveals its value. Dividing both sides by the same non-zero number maintains the equation's balance and ensures that the solution remains accurate. This simple yet crucial step completes the process of solving for x, providing the numerical value that satisfies the original equation. The ability to isolate a variable through division is a fundamental skill in algebra and is essential for solving a wide range of equations. By performing this final operation, we unveil the solution to the equation, demonstrating the power of algebraic manipulation in determining unknown quantities.

Divide both sides by 7:

rac{14}{7} = rac{7x}{7}

This gives us the solution:

$x = 2$

Verification: Ensuring the Accuracy of Our Solution

After arriving at a solution, it is paramount to verify its accuracy. This crucial step involves substituting the obtained value of x back into the original equation. If the solution is correct, both sides of the equation will evaluate to the same value, confirming the validity of our result. This verification process serves as a safeguard against potential errors made during the equation-solving process. Errors can arise from various sources, such as incorrect distribution, combination of like terms, or arithmetic mistakes. By substituting the solution back into the original equation, we can detect and rectify these errors. This practice not only ensures the accuracy of the solution but also reinforces the understanding of the equation's structure and the relationships between its terms. The verification step instills confidence in the obtained solution and demonstrates a thorough approach to problem-solving. It is an integral part of the mathematical process and should never be overlooked. Let's substitute $x = 2$ into the original equation.

Original Equation: $-\frac{1}{2}(-10 x-10)-8=4(3 x-2)-9$

Substitute $x = 2$:

$-\frac{1}{2}(-10(2)-10)-8=4(3(2)-2)-9$

Simplify the left side:

$-\frac{1}{2}(-20-10)-8$

$-\frac{1}{2}(-30)-8$

$15 - 8 = 7$

Simplify the right side:

$4(6-2)-9$

$4(4)-9$

$16 - 9 = 7$

Since both sides of the equation simplify to 7, our solution $x = 2$ is verified to be correct.

Conclusion

In this comprehensive guide, we have meticulously dissected the process of solving the linear equation $-\frac{1}{2}(-10 x-10)-8=4(3 x-2)-9$. We embarked on this mathematical journey by first distributing constants to eliminate parentheses, a crucial step in simplifying the equation. We then combined like terms on both sides, streamlining the equation and making it more manageable. Next, we strategically isolated the variable term, moving all x terms to one side and constants to the other. This paved the way for the final step of solving for x, where we divided both sides by the coefficient of x, revealing the solution. To ensure the accuracy of our result, we emphasized the importance of verification, substituting the obtained value back into the original equation. This process not only confirmed the correctness of our solution but also reinforced our understanding of the equation's structure. The solution, $x = 2$, was rigorously verified, solidifying our confidence in the result. This comprehensive approach to solving linear equations provides a framework for tackling similar problems with clarity and precision. By mastering these techniques, learners can confidently navigate the world of algebra and apply these skills to various real-world scenarios. The ability to solve linear equations is a cornerstone of mathematical literacy, empowering individuals to analyze problems, make informed decisions, and approach challenges with a logical and systematic mindset. As we conclude this guide, we encourage you to continue practicing and exploring the fascinating world of mathematics, where every equation solved is a step forward in understanding the underlying principles that govern our world.