Solving The Equation 12y = 132 A Step-by-Step Guide

Understanding the Basics of Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They often involve variables, which are symbols representing unknown values. In the equation 12y = 132, 'y' is the variable we aim to find. The goal is to isolate the variable on one side of the equation to determine its value. To achieve this, we use inverse operations, which are operations that undo each other. For instance, the inverse operation of multiplication is division, and vice versa. Understanding these basic principles is fundamental to solving any algebraic equation. In this particular case, the equation is a linear equation in one variable, which means it involves a single variable raised to the power of one. These types of equations are the building blocks of more complex algebraic problems, and mastering their solution is essential for further mathematical studies. Furthermore, the concept of equality is paramount; whatever operation is performed on one side of the equation must also be performed on the other side to maintain the balance and ensure the solution remains valid.

Step-by-Step Solution of 12y = 132

To solve the equation 12y = 132, we need to isolate the variable 'y'. This means we want to get 'y' by itself on one side of the equation. Since 'y' is being multiplied by 12, we need to perform the inverse operation, which is division. We divide both sides of the equation by 12 to maintain the balance and ensure the equation remains true. This step is crucial because it applies the fundamental principle of equality in algebra, which states that any operation performed on one side of the equation must be performed on the other side. By dividing both sides by 12, we effectively undo the multiplication by 12 on the left side, leaving 'y' isolated. This process simplifies the equation and brings us closer to finding the value of 'y'. The arithmetic involved is straightforward: dividing 132 by 12 yields 11. This result provides the solution to the equation, demonstrating the value of 'y' that satisfies the original statement.

Step 1: Isolate the Variable

The equation is 12y = 132. To isolate 'y', we divide both sides of the equation by 12. This gives us:

12y / 12 = 132 / 12

Step 2: Perform the Division

Performing the division, we get:

y = 11

Therefore, the solution to the equation 12y = 132 is y = 11. This step is the culmination of the isolation process, where the variable 'y' is now completely by itself on one side of the equation. The result, y = 11, represents the value that, when substituted back into the original equation, will make the equation true. The simplicity of this final step underscores the effectiveness of the initial strategy of isolating the variable through inverse operations. It highlights the power of algebraic manipulation in simplifying complex problems into manageable steps. Moreover, it reinforces the importance of accurate arithmetic in arriving at the correct solution.

Verification of the Solution

Verification is a crucial step in solving any equation. To verify our solution, we substitute the value we found for 'y' back into the original equation. If the equation holds true, then our solution is correct. In this case, we substitute y = 11 into the equation 12y = 132. This process of substitution is essential because it confirms that the value we obtained for 'y' indeed satisfies the conditions of the original equation. It acts as a safeguard against potential errors made during the solution process. By performing this verification, we ensure the accuracy and validity of our answer. This step not only provides confidence in the solution but also reinforces the understanding of the relationship between the variable and the equation. It emphasizes that the solution is not just a numerical value but a value that makes the equation a true statement.

Substituting y = 11 into the Original Equation

Substituting y = 11 into 12y = 132, we get:

12 * 11 = 132

Checking the Equality

Since 12 * 11 = 132, the equation holds true. This confirms that our solution y = 11 is correct. This final check solidifies the correctness of the solution, providing a sense of completion and assurance. It demonstrates the practical application of the algebraic principles used in solving the equation. The fact that the left-hand side of the equation equals the right-hand side when y = 11 confirms that we have successfully found the value of 'y' that satisfies the given condition. This verification process is a valuable skill in mathematics, as it allows one to independently check the accuracy of their work and build confidence in their problem-solving abilities.

Why Understanding Algebraic Equations is Important

Understanding algebraic equations is fundamental to many areas of mathematics and real-world applications. Algebra is the language of mathematics, and equations are its sentences. Being able to solve equations is essential for understanding more advanced mathematical concepts, such as calculus, linear algebra, and differential equations. Beyond mathematics, algebraic skills are crucial in various fields, including science, engineering, economics, and computer science. For instance, in physics, equations are used to describe the laws of motion; in economics, they are used to model market behavior; and in computer science, they are used in algorithms and programming. The ability to manipulate and solve equations allows professionals to analyze data, make predictions, and solve problems effectively. Moreover, the logical thinking and problem-solving skills developed through algebra are transferable to many other aspects of life, enhancing one's ability to approach challenges systematically and find solutions. Therefore, mastering algebraic equations is not just an academic exercise but a valuable life skill that opens doors to numerous opportunities.

Common Mistakes to Avoid When Solving Equations

When solving equations, it's easy to make mistakes if you're not careful. One common mistake is not performing the same operation on both sides of the equation. Remember, the equation must remain balanced. Another mistake is misapplying the order of operations. Always follow the correct order (PEMDAS/BODMAS) to avoid errors. Sign errors are also common, especially when dealing with negative numbers. Take extra care when adding, subtracting, multiplying, or dividing negative numbers. Finally, always verify your solution by substituting it back into the original equation. This will help you catch any mistakes you might have made. Being aware of these common pitfalls and taking steps to avoid them can significantly improve your accuracy and confidence in solving algebraic equations. It's also helpful to practice regularly and seek clarification when you're unsure about a concept or step. With consistent effort and attention to detail, you can minimize errors and become proficient in solving equations.

Conclusion: Mastering Algebraic Equations

In conclusion, solving the equation 12y = 132 demonstrates a fundamental algebraic principle: isolating the variable using inverse operations. The step-by-step process involved dividing both sides of the equation by 12, resulting in the solution y = 11. We then verified this solution by substituting it back into the original equation, confirming its correctness. This exercise highlights the importance of understanding algebraic equations and the techniques used to solve them. Mastering these skills is crucial for success in higher-level mathematics and various real-world applications. By understanding the underlying principles and practicing regularly, you can build confidence in your ability to solve algebraic equations and tackle more complex problems. Remember, the key to success in mathematics is consistent effort, attention to detail, and a willingness to learn from mistakes. With these qualities, you can unlock the power of algebra and its applications in numerous fields.