Solving Linear Equations A Step-by-Step Guide To -9h - 6 + 12h + 40 = 22
Linear equations are the cornerstone of algebra and form the basis for more advanced mathematical concepts. Mastering the techniques to solve linear equations is crucial for success in mathematics and various fields that rely on quantitative analysis. In this article, we will delve into the step-by-step process of solving the linear equation -9h - 6 + 12h + 40 = 22, providing a clear and comprehensive guide for students and anyone seeking to enhance their algebraic skills. Understanding the fundamental principles and applying them systematically will empower you to tackle a wide range of linear equations with confidence. This article aims to provide a detailed explanation of each step involved in solving the equation, ensuring that you grasp the underlying concepts and can apply them to similar problems.
Understanding Linear Equations
Before we dive into solving the equation, it's essential to understand what a linear equation is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variable is raised to the power of one, and the equation can be graphically represented as a straight line. Linear equations are characterized by their simplicity and predictability, making them a fundamental topic in algebra. The general form of a linear equation in one variable is ax + b = c, where a, b, and c are constants, and x is the variable. In our case, the equation -9h - 6 + 12h + 40 = 22 fits this description, with h being the variable we need to solve for. Recognizing the structure of a linear equation is the first step towards finding its solution. Linear equations can also appear in more complex forms, but the core principle remains the same: to isolate the variable on one side of the equation and find its value. This involves using algebraic manipulations to simplify the equation while maintaining its balance. Understanding the properties of equality, such as the addition and multiplication properties, is crucial for solving linear equations effectively. These properties allow us to perform operations on both sides of the equation without changing its solution.
Step 1: Combine Like Terms
The first step in solving the equation -9h - 6 + 12h + 40 = 22 is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this equation, we have two terms with the variable h: -9h and 12h. We also have two constant terms: -6 and 40. Combining like terms simplifies the equation, making it easier to solve. To combine the terms with h, we add their coefficients: -9 + 12 = 3. So, -9h + 12h simplifies to 3h. Next, we combine the constant terms: -6 + 40 = 34. Therefore, the left side of the equation simplifies to 3h + 34. Now, the equation looks much simpler: 3h + 34 = 22. This step is crucial because it reduces the number of terms and makes the equation more manageable. Combining like terms is a fundamental algebraic technique that is used in various mathematical contexts. It helps to organize and simplify expressions, making them easier to work with. By combining like terms, we reduce the complexity of the equation, making it easier to isolate the variable and find its value. This step is a prerequisite for many algebraic manipulations and is essential for solving equations efficiently.
Step 2: Isolate the Variable Term
After combining like terms, the equation is now 3h + 34 = 22. The next step is to isolate the variable term, which means getting the term with the variable (3h in this case) alone on one side of the equation. To do this, we need to eliminate the constant term on the same side as the variable term. In this equation, the constant term is 34. To eliminate it, we subtract 34 from both sides of the equation. This is based on the subtraction property of equality, which states that if we subtract the same number from both sides of an equation, the equation remains balanced. So, we perform the operation: 3h + 34 - 34 = 22 - 34. This simplifies to 3h = -12. Subtracting 34 from both sides effectively isolates the variable term, bringing us closer to solving for h. Isolating the variable term is a critical step in solving linear equations because it sets the stage for the final step of finding the variable's value. This step involves using inverse operations to undo the operations that are being performed on the variable. In this case, we used subtraction to undo addition. The goal is to simplify the equation until the variable term is the only term on one side of the equation. This makes it clear what operation needs to be performed to solve for the variable.
Step 3: Solve for the Variable
Now that we have isolated the variable term, the equation is 3h = -12. The final step is to solve for the variable h. To do this, we need to undo the multiplication that is being performed on h. In this case, h is being multiplied by 3. To undo this multiplication, we divide both sides of the equation by 3. This is based on the division property of equality, which states that if we divide both sides of an equation by the same non-zero number, the equation remains balanced. So, we perform the operation: (3h) / 3 = (-12) / 3. This simplifies to h = -4. Therefore, the solution to the equation -9h - 6 + 12h + 40 = 22 is h = -4. Solving for the variable is the ultimate goal of solving any equation. This step involves using inverse operations to isolate the variable and find its value. In this case, we used division to undo multiplication. The solution, h = -4, represents the value that makes the original equation true. To verify this solution, we can substitute -4 back into the original equation and check if both sides are equal. This process of verification ensures that the solution is correct and that no errors were made during the solving process.
Step 4: Verify the Solution
To ensure the accuracy of our solution, it's crucial to verify the solution. We found that h = -4, so we will substitute this value back into the original equation: -9h - 6 + 12h + 40 = 22. Substituting h = -4, we get: -9(-4) - 6 + 12(-4) + 40 = 22. Now, we perform the arithmetic operations: 36 - 6 - 48 + 40 = 22. Simplifying further: 30 - 48 + 40 = 22. And then: -18 + 40 = 22. Finally, we have: 22 = 22. Since both sides of the equation are equal, our solution h = -4 is correct. Verifying the solution is an essential step in the problem-solving process. It helps to catch any errors that may have been made during the algebraic manipulations. By substituting the solution back into the original equation, we can confirm whether it satisfies the equation. If the two sides of the equation are equal, then the solution is correct. If they are not equal, it indicates that there was an error somewhere in the solving process, and we need to go back and review each step to find the mistake. Verification not only ensures the correctness of the solution but also reinforces the understanding of the equation and the steps involved in solving it.
Conclusion
In this article, we have provided a comprehensive guide to solving the linear equation -9h - 6 + 12h + 40 = 22. We have demonstrated the importance of combining like terms, isolating the variable term, and solving for the variable. Additionally, we emphasized the crucial step of verifying the solution to ensure its accuracy. By following these steps carefully, you can confidently solve a wide range of linear equations. Mastering these techniques will not only enhance your algebraic skills but also provide a solid foundation for more advanced mathematical concepts. Linear equations are a fundamental part of mathematics, and the ability to solve them is essential for success in various fields. The step-by-step approach outlined in this article provides a clear and systematic method for solving linear equations, making it accessible to students and anyone looking to improve their mathematical abilities. By practicing these techniques and applying them to different problems, you can develop a strong understanding of linear equations and their solutions. The key to success is consistent practice and attention to detail.
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