Solving Linear Equations A Step-by-Step Guide To Simplifying And Finding X

In the realm of mathematics, solving linear equations is a fundamental skill. Linear equations, characterized by variables raised to the first power, appear in various mathematical and real-world scenarios. This article delves into the process of solving a specific linear equation, providing a comprehensive step-by-step guide to simplify the equation and isolate the variable x. By understanding the underlying principles and applying the correct techniques, you can confidently tackle linear equations of varying complexity. This detailed explanation aims to equip you with the knowledge and skills necessary to confidently approach and solve similar problems. Linear equations are not just theoretical exercises; they are essential tools for modeling and understanding relationships in diverse fields, from physics and engineering to economics and finance. Mastering the art of solving them opens doors to a deeper appreciation of mathematical problem-solving and its practical applications.

The given equation is:

$$-10 x+10(-10 x-6)=-6(10 x-1)-5$$

This equation might seem daunting at first glance, but by systematically applying the principles of algebra, we can simplify it and arrive at the solution. The key to solving any equation lies in maintaining balance – performing the same operations on both sides to ensure that the equality remains valid. We'll start by expanding the expressions, combining like terms, and then strategically isolating the variable x to find its value. This process involves a series of steps, each building upon the previous one, to progressively simplify the equation until the solution becomes clear. Throughout this journey, it's crucial to pay attention to detail, ensuring that each step is performed accurately. A single mistake can lead to an incorrect answer, so careful attention and methodical application of the rules of algebra are paramount. By breaking down the problem into manageable steps and meticulously executing each one, we can successfully navigate the complexities of the equation and arrive at the correct solution. The satisfaction of solving a challenging equation lies not just in finding the answer but also in the process of unraveling the complexities and understanding the underlying mathematical principles.

Step 1: Distribute

The first step in simplifying the equation is to distribute the constants outside the parentheses to the terms inside. This process removes the parentheses, making it easier to combine like terms and isolate the variable. Distribution involves multiplying the constant outside the parentheses by each term inside the parentheses. This is a fundamental algebraic operation based on the distributive property, which states that a(b + c) = ab + ac. Applying this property correctly is crucial for accurately simplifying the equation. Failing to distribute correctly can lead to errors that propagate through the rest of the solution, ultimately resulting in a wrong answer. Therefore, it's important to be meticulous and ensure that each term inside the parentheses is multiplied by the constant outside. This seemingly simple step lays the foundation for the subsequent steps, making the equation more manageable and paving the way for a successful solution.

Applying the distributive property, we get:

$$-10x + 10(-10x) + 10(-6) = -6(10x) -6(-1) - 5$$

Simplifying each term:

$$-10x - 100x - 60 = -60x + 6 - 5$$

Step 2: Combine Like Terms

Now that we've eliminated the parentheses, the next step is to combine like terms on each side of the equation. Like terms are those that have the same variable raised to the same power (or are constants). Combining them simplifies the equation by reducing the number of terms and making it easier to isolate the variable. This process involves adding or subtracting the coefficients of like terms. For example, 2x and 3x are like terms and can be combined to get 5x. Similarly, 5 and -2 are like terms and can be combined to get 3. Combining like terms is a crucial step in solving equations because it streamlines the equation and brings us closer to the solution. Neglecting to combine like terms can lead to unnecessary complexity and make it more difficult to isolate the variable. Therefore, it's important to carefully identify and combine like terms on each side of the equation before proceeding to the next step. This will ensure a smoother and more efficient solution process.

On the left side, we combine the x terms: -10x and -100x. On the right side, we combine the constants 6 and -5.

$$-110x - 60 = -60x + 1$$

Step 3: Isolate the Variable Term

The goal now is to isolate the term containing the variable x on one side of the equation. This means moving all terms with x to one side and all constant terms to the other side. To do this, we use inverse operations. Inverse operations are operations that undo each other, such as addition and subtraction, or multiplication and division. By adding or subtracting the same term from both sides of the equation, we can move terms from one side to the other without changing the equation's balance. This is a crucial step in solving for x because it brings all the x terms together, allowing us to eventually isolate x itself. The strategic use of inverse operations is a fundamental technique in algebra and is essential for solving a wide range of equations. Choosing the correct inverse operation and applying it consistently to both sides of the equation is key to maintaining balance and progressing towards the solution.

To isolate the x term, we add 60x to both sides of the equation:

$$-110x - 60 + 60x = -60x + 1 + 60x$$

Simplifying:

$$-50x - 60 = 1$$

Next, add 60 to both sides to isolate the term with x:

$$-50x - 60 + 60 = 1 + 60$$

Simplifying:

$$-50x = 61$$

Step 4: Solve for x

Finally, to solve for x, we need to isolate x completely. This involves dividing both sides of the equation by the coefficient of x. The coefficient of x is the number that is multiplied by x. In this case, the coefficient of x is -50. Dividing both sides of the equation by the coefficient of x is the inverse operation of multiplication and will leave x isolated on one side of the equation. This is the final step in solving for x and will give us the value of x that satisfies the original equation. It's important to remember that when dividing by a negative number, the sign of the resulting quotient will be the opposite of the sign of the dividend. Paying attention to signs is crucial for obtaining the correct solution.

Divide both sides by -50:

$$\frac{-50x}{-50} = \frac{61}{-50}$$

Simplifying, we get:

$$x = -\frac{61}{50}$$

Step 5: Express the Answer in the Simplest Form

The solution we obtained is x = -61/50. It's crucial to express the answer in the simplest form. In this case, the fraction is already in its simplest form because 61 and 50 have no common factors other than 1. However, if the fraction could be further simplified, we would need to divide both the numerator and denominator by their greatest common divisor. Expressing the answer in the simplest form ensures that the solution is presented in the most concise and easily understandable manner. It also reflects a good mathematical practice of presenting results in their most reduced form. In some cases, the answer may also be expressed as a mixed number or a decimal, depending on the context of the problem. However, for many algebraic problems, leaving the answer as a simplified fraction is the preferred format.

Therefore, the simplest form of the solution is:

$$x = -\frac{61}{50}$$

Conclusion

In this article, we have demonstrated a detailed, step-by-step approach to solving a linear equation. We began by distributing constants, combined like terms, isolated the variable term, and finally solved for x. Each step was carefully explained to provide a clear understanding of the process. Solving linear equations is a fundamental skill in mathematics, and mastering this process can pave the way for solving more complex problems. The ability to manipulate equations, isolate variables, and simplify expressions is essential for success in algebra and beyond. By practicing these techniques and understanding the underlying principles, you can develop confidence in your problem-solving abilities and tackle a wide range of mathematical challenges. The journey of solving an equation is not just about finding the answer; it's about developing a logical and systematic approach to problem-solving, a skill that is valuable in many aspects of life. The satisfaction of successfully solving a challenging equation comes from the understanding that you have mastered a powerful tool for analyzing and solving problems.

In summary, the solution to the equation

$$-10 x+10(-10 x-6)=-6(10 x-1)-5$$

is:

$$x = -\frac{61}{50}$$

This solution represents the value of x that makes the equation true. By substituting this value back into the original equation, we can verify that both sides of the equation are equal. This process of verification is a good practice to ensure that the solution is correct. It also reinforces the understanding of what it means to solve an equation – to find the value(s) of the variable(s) that make the equation a true statement. The solution we have found is not just a number; it's a key that unlocks the relationship expressed by the equation. It's a testament to the power of algebra to represent and solve real-world problems.