Solving For X In (4x + 3)° + (x - 8)° = 90°

Introduction: The Importance of Understanding Algebraic Equations

In the realm of mathematics, specifically in algebra, solving equations is a fundamental skill. It's not just about finding the value of an unknown variable; it's about developing logical thinking, problem-solving strategies, and a deeper understanding of how mathematical relationships work. In this article, we'll embark on a journey to solve the equation (4x + 3)° + (x - 8)° = 90°, breaking down each step and providing a clear explanation to help you grasp the underlying concepts. Before diving into the solution, let's emphasize the significance of algebraic equations in various fields. From engineering and physics to economics and computer science, equations are the building blocks for modeling real-world phenomena and making predictions. Mastering the art of solving equations opens doors to a wide array of career paths and enhances your ability to tackle complex problems in everyday life. This article is designed to be more than just a step-by-step solution; it's a guide that empowers you to approach similar problems with confidence and a solid understanding of the mathematical principles involved.

Problem Statement: Decoding the Equation (4x + 3)° + (x - 8)° = 90°

Let's begin by carefully examining the equation we aim to solve: (4x + 3)° + (x - 8)° = 90°. This equation involves the variable x, which represents an unknown value we need to determine. The expressions (4x + 3)° and (x - 8)° represent angles measured in degrees. The core of the problem lies in understanding the relationship between these angles and the fact that their sum equals 90 degrees. This suggests that we're dealing with complementary angles, a concept we'll revisit later. Understanding the problem statement is crucial before attempting to solve any equation. We need to identify the unknowns, the given information, and the relationships between them. In this case, the unknown is x, the given information is the equation itself, and the relationship is the sum of the two angle expressions equaling 90 degrees. To effectively tackle this problem, we'll need to employ algebraic manipulation techniques, such as combining like terms, isolating the variable, and performing arithmetic operations. We'll also need to keep in mind the properties of angles and their relationships. By carefully dissecting the problem statement, we can formulate a strategic approach to find the value of x. This problem not only tests our algebraic skills but also our ability to interpret mathematical notation and apply geometric concepts.

Step-by-Step Solution: Unraveling the Value of x

Now, let's embark on the journey of solving the equation (4x + 3)° + (x - 8)° = 90° step by step. Our primary goal is to isolate x on one side of the equation to determine its value. The first step is to remove the parentheses. Since we are adding the two expressions, we can simply rewrite the equation as: 4x + 3 + x - 8 = 90. This step utilizes the associative property of addition, which allows us to group the terms without changing the outcome. Next, we need to combine like terms. Like terms are those that have the same variable raised to the same power. In this case, 4x and x are like terms, and 3 and -8 are like terms. Combining these terms, we get: 5x - 5 = 90. This step simplifies the equation, making it easier to work with. The following step involves isolating the term with x. To do this, we add 5 to both sides of the equation: 5x - 5 + 5 = 90 + 5, which simplifies to 5x = 95. This step maintains the equality of the equation while moving the constant term to the right side. Finally, to solve for x, we divide both sides of the equation by 5: 5x / 5 = 95 / 5, which gives us x = 19. This is the solution to the equation. We have successfully isolated x and found its value. It's essential to double-check our solution by substituting x = 19 back into the original equation to ensure it holds true. This step verifies the accuracy of our calculations and provides confidence in our answer.

Verification: Ensuring the Accuracy of Our Solution

To ensure the accuracy of our solution, x = 19, we need to substitute this value back into the original equation (4x + 3)° + (x - 8)° = 90°. Substituting x = 19, we get: (4 * 19 + 3)° + (19 - 8)° = 90°. Now, let's simplify each expression. First, 4 * 19 = 76, so (4 * 19 + 3)° becomes (76 + 3)° = 79°. Next, (19 - 8)° = 11°. Therefore, our equation now reads: 79° + 11° = 90°. Adding the angles, we have 79 + 11 = 90, which confirms that 90° = 90°. This equality demonstrates that our solution, x = 19, is indeed correct. Verification is a crucial step in problem-solving, especially in mathematics. It allows us to catch any errors in our calculations and ensures that our answer satisfies the conditions of the problem. By substituting the solution back into the original equation, we can gain confidence in our result and avoid potential mistakes. In this case, the verification process has not only confirmed the accuracy of our solution but also reinforced our understanding of the equation and the relationships between its components. This step is a testament to the importance of thoroughness and attention to detail in mathematical problem-solving.

Conclusion: The Power of Algebraic Problem Solving

In conclusion, we have successfully solved the equation (4x + 3)° + (x - 8)° = 90° and found the value of x to be 19. This journey has highlighted the importance of understanding algebraic concepts, applying problem-solving strategies, and verifying the accuracy of our solutions. The ability to solve equations like this is a fundamental skill in mathematics and has wide-ranging applications in various fields. We began by carefully dissecting the problem statement, identifying the unknown variable, and recognizing the relationship between the angle expressions. We then proceeded with a step-by-step solution, employing algebraic manipulation techniques such as combining like terms, isolating the variable, and performing arithmetic operations. The key to success was to break down the problem into smaller, manageable steps and to maintain clarity and precision in our calculations. Finally, we verified our solution by substituting x = 19 back into the original equation, ensuring that it satisfied the given conditions. This process not only confirmed the accuracy of our answer but also reinforced our understanding of the equation and the underlying mathematical principles. The ability to solve algebraic equations is not just about finding a numerical answer; it's about developing critical thinking skills, logical reasoning, and the ability to approach complex problems with confidence. These skills are valuable assets in any academic or professional endeavor, making the mastery of algebra a worthwhile pursuit.