Solving 9(-9x + 5) + 10 = 2(-10x + 2) - 3 A Step-by-Step Guide

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In mathematics, solving equations is a fundamental skill. Linear equations, in particular, are the building blocks for more advanced concepts. This comprehensive guide provides a detailed, step-by-step approach to solving the linear equation 9(-9x + 5) + 10 = 2(-10x + 2) - 3 for x, ensuring clarity and understanding at each stage. Our goal is to express the answer in its simplest form, a crucial aspect of mathematical problem-solving. This article aims to break down the equation-solving process, making it accessible to students and anyone looking to refresh their algebra skills. The ability to manipulate and solve equations is essential in various fields, including science, engineering, economics, and computer science. Therefore, mastering this skill is invaluable for both academic and practical applications. Let's embark on this journey to understand the intricacies of linear equations and discover the value of 'x' in the given problem.

1. Distribute and Simplify: The Foundation of Equation Solving

The first crucial step in solving the equation 9(-9x + 5) + 10 = 2(-10x + 2) - 3 involves distributing the numbers outside the parentheses to the terms inside. This process, based on the distributive property of multiplication over addition and subtraction, allows us to eliminate the parentheses and create a more manageable equation. Let's break it down:

  • Distribute the 9: Multiply 9 by both -9x and +5. This gives us 9 * -9x = -81x and 9 * 5 = 45. So, the left side of the equation becomes -81x + 45.
  • Distribute the 2: On the right side, multiply 2 by both -10x and +2. This yields 2 * -10x = -20x and 2 * 2 = 4. Thus, the right side of the equation transforms into -20x + 4.

After distributing, our equation now looks like this: -81x + 45 + 10 = -20x + 4 - 3. The next part is simplify both sides by combining constant terms. On the left side, we have +45 + 10, which simplifies to +55. On the right side, we have +4 - 3, which simplifies to +1. So, the equation further simplifies to -81x + 55 = -20x + 1. This simplified form makes it easier to isolate the variable x and move towards finding its value. This initial step of distribution and simplification is not just about making the equation look simpler; it's about transforming it into a form where we can apply further algebraic techniques effectively. By removing parentheses and combining like terms, we pave the way for the subsequent steps of isolating the variable and ultimately solving the equation.

2. Isolate the Variable Terms: Bringing 'x' to One Side

After simplifying the equation to -81x + 55 = -20x + 1, the next critical step is to isolate the terms containing the variable x on one side of the equation. This is achieved by strategically adding or subtracting terms from both sides, maintaining the equation's balance. The goal here is to group all the x terms together, making it easier to eventually solve for x. To accomplish this, we can add 81x to both sides of the equation. This eliminates the -81x term on the left side and moves the x term to the right side. Adding 81x to both sides, we get: -81x + 55 + 81x = -20x + 1 + 81x. Simplifying this gives us: 55 = 61x + 1. Now, all the x terms are on the right side of the equation. The next substep is subtract 1 from both sides of the equation. By subtracting 1 from both sides, we isolate the x term further. Subtracting 1 from both sides, we have: 55 - 1 = 61x + 1 - 1. This simplifies to 54 = 61x. At this stage, we have successfully isolated the x term on one side of the equation, which is a significant milestone in solving for x. The equation 54 = 61x is now in a form where we can easily find the value of x by performing a single operation. This process of isolating the variable terms is a fundamental technique in algebra, applicable to a wide range of equation-solving scenarios. By carefully manipulating the equation while maintaining its balance, we bring ourselves closer to the solution.

3. Solve for 'x': The Final Calculation

Having successfully isolated the variable term, our equation now stands as 54 = 61x. The final step in solving for x is to isolate x completely. This is achieved by performing the inverse operation of what is currently being done to x. In this case, x is being multiplied by 61. Therefore, to isolate x, we need to divide both sides of the equation by 61. Dividing both sides by 61, we get: 54 / 61 = (61x) / 61. On the right side, 61 divided by 61 equals 1, so we are left with just x. Therefore, the equation simplifies to: x = 54/61. This fraction, 54/61, represents the value of x that satisfies the original equation. Now, it is time to consider if the fraction can be simplified further. To simplify a fraction, we look for common factors between the numerator (the top number) and the denominator (the bottom number). In this case, 54 and 61 do not share any common factors other than 1. This means that the fraction 54/61 is already in its simplest form. Therefore, the final solution for x is 54/61. This step, where we perform the final calculation to find the value of the variable, is the culmination of all the previous steps. It demonstrates the power of algebraic manipulation in solving equations. By carefully applying the rules of algebra, we have successfully navigated through the equation and arrived at the solution. This process not only gives us the answer but also reinforces the understanding of the underlying mathematical principles.

4. Verification: Ensuring Accuracy in Your Solution

After solving for x and obtaining the solution x = 54/61, it's crucial to verify the accuracy of our result. Verification involves substituting the calculated value of x back into the original equation to ensure that both sides of the equation are equal. This step is an essential practice in mathematics, as it helps to catch any potential errors made during the solving process. It provides confidence in the correctness of the solution and reinforces the understanding of the equation's properties. Let's substitute x = 54/61 into the original equation: 9(-9x + 5) + 10 = 2(-10x + 2) - 3. Replacing x with 54/61, we get: 9(-9(54/61) + 5) + 10 = 2(-10(54/61) + 2) - 3. Now, we simplify each side of the equation separately.

Left Side Simplification:

  1. Calculate -9(54/61): This equals -486/61.
  2. Add 5: Convert 5 to a fraction with a denominator of 61 (5 = 305/61). So, -486/61 + 305/61 = -181/61.
  3. Multiply by 9: 9 * (-181/61) = -1629/61.
  4. Add 10: Convert 10 to a fraction with a denominator of 61 (10 = 610/61). So, -1629/61 + 610/61 = -1019/61.

Right Side Simplification:

  1. Calculate -10(54/61): This equals -540/61.
  2. Add 2: Convert 2 to a fraction with a denominator of 61 (2 = 122/61). So, -540/61 + 122/61 = -418/61.
  3. Multiply by 2: 2 * (-418/61) = -836/61.
  4. Subtract 3: Convert 3 to a fraction with a denominator of 61 (3 = 183/61). So, -836/61 - 183/61 = -1019/61.

Comparing both sides, we see that the left side (-1019/61) is equal to the right side (-1019/61). This confirms that our solution, x = 54/61, is correct. The verification process is more than just a check; it's a way to solidify understanding and build confidence in one's problem-solving abilities. It reinforces the interconnectedness of mathematical operations and the importance of accuracy in each step.

Conclusion: Mastering Linear Equations

In summary, we have successfully solved the linear equation 9(-9x + 5) + 10 = 2(-10x + 2) - 3 for x, arriving at the solution x = 54/61. This journey involved several key steps: distribution and simplification, isolating the variable terms, solving for x, and finally, verifying the solution. Each step is crucial in the process of solving linear equations, and mastering these steps is fundamental to success in algebra and beyond. We began by distributing the constants outside the parentheses to the terms inside, effectively eliminating the parentheses and simplifying the equation. This step transformed the equation into a more manageable form, setting the stage for further algebraic manipulation. Next, we isolated the variable terms on one side of the equation. This involved strategically adding or subtracting terms from both sides to group all the x terms together. This step is essential for isolating the variable and paving the way for solving it. Then, we solved for x by performing the inverse operation of what was being done to x. In this case, we divided both sides of the equation by the coefficient of x, which gave us the value of x. Finally, we verified our solution by substituting it back into the original equation. This step is crucial for ensuring the accuracy of our solution and catching any potential errors. The successful completion of this process demonstrates a strong understanding of linear equations and the algebraic techniques required to solve them. The ability to solve linear equations is a valuable skill that extends beyond the classroom, with applications in various fields such as science, engineering, economics, and computer science. By mastering these fundamental concepts, you build a strong foundation for tackling more complex mathematical problems in the future. Remember, practice is key to mastering any mathematical skill. The more you practice solving linear equations, the more confident and proficient you will become. Embrace the challenges, learn from your mistakes, and celebrate your successes. With consistent effort and a solid understanding of the underlying principles, you can unlock the power of algebra and excel in your mathematical journey.