Solving For 'c': The Equation 12c - 53 = 91

Solving for 'c': The Equation 12c - 53 = 91

When you're faced with an algebraic equation like $12c - 53 = 91$, the main goal is to isolate the variable, in this case, '$c$', to find its unknown value. This process involves a series of logical steps, applying inverse operations to both sides of the equation to maintain balance. Think of an equation as a perfectly balanced scale. Whatever you do to one side, you must do to the other to keep it level. Our target is to get '$c$' all by itself on one side of the '=' sign. The number 53 is currently being subtracted from $12c$. To undo subtraction, we use its opposite operation: addition. So, the first strategic move is to add 53 to both sides of the equation. This will cancel out the '-53' on the left side, bringing us one step closer to isolating '$c$'. Remember, balance is key, so we add 53 to $91$ on the right side as well. This step simplifies the equation significantly, making it easier to tackle the next stage of the solution process. The aim is always to simplify and move towards the isolation of the variable, and this initial addition is a crucial maneuver in achieving that goal. By performing this addition, we are effectively removing the constant term that is interfering with the '$c$' term, preparing the equation for the final step of determining the value of '$c$'. This methodical approach ensures that no errors are made and that the integrity of the original equation is preserved throughout the solving process. It’s all about undoing operations step-by-step, like peeling an onion, to get to the core value.

Once we've added 53 to both sides, our equation transforms. On the left side, $-53 + 53$ becomes 0, leaving us with just $12c$. On the right side, $91 + 53$ gives us a new total. Let's do the addition: $91 + 50 = 141$, and then $141 + 3 = 144$. So, the equation now reads $12c = 144$. We're much closer to finding the value of '$c$'! Now, '$c$' is being multiplied by 12. To isolate '$c$', we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the equation by 12. This will cancel out the 12 that's multiplying '$c$' on the left side, leaving '$c$' standing alone. On the right side, we'll divide 144 by 12. This division will give us the final numerical value for '$c$'. Performing this division, $144 \/ 12$, we find that $c = 12$. This is our solution. It's essential to double-check our answer by substituting $c = 12$ back into the original equation: $12(12) - 53$. First, calculate $12 \times 12$, which is 144. Then, subtract 53: $144 - 53 = 91$. Since $91$ is equal to the right side of the original equation, our solution $c=12$ is correct. This verification step is a critical part of the problem-solving process, as it confirms the accuracy of our calculations and ensures that we have indeed found the correct value for the variable. It provides confidence in the result and reinforces understanding of the algebraic principles at play.

Therefore, in the equation $12c - 53 = 91$, the value of $c$ is 12. This corresponds to option C. It's a satisfying moment when you arrive at the solution and can confirm its correctness. This type of problem is fundamental in algebra and appears in many contexts, from basic arithmetic exercises to more complex scientific and engineering applications. Mastering these foundational skills is key to unlocking further mathematical understanding. The process of isolating a variable through inverse operations is a cornerstone of algebraic manipulation. It teaches us to think systematically and to apply logical reasoning to solve for unknowns. Each step, from adding to both sides to dividing both sides, is a deliberate move designed to simplify the equation and bring us closer to the answer. The satisfaction comes not just from finding the number, but from understanding the 'why' behind each step. It’s about building a mental toolkit that can be applied to a vast array of problems. Remember, algebra isn't just about numbers; it's about patterns, relationships, and logical deduction. The journey to solve for '$c$' in $12c - 53 = 91$ is a mini-masterclass in these principles, demonstrating how consistent application of rules leads to a definitive answer. The process is elegant in its simplicity and powerful in its application. It highlights how seemingly complex problems can be broken down into manageable steps, each building upon the last. The core idea is always to maintain equality while systematically isolating the variable. This principle extends far beyond this single equation, forming the basis for solving systems of equations, inequalities, and many other advanced mathematical concepts. So, while finding the value of '$c$' is the immediate goal, the real takeaway is the mastery of a fundamental problem-solving technique.

For further exploration into algebraic equations and problem-solving techniques, you can visit the Khan Academy website. They offer a wealth of free resources and tutorials that can help you deepen your understanding of mathematics, from basic arithmetic to calculus. Their interactive exercises and clear explanations make learning engaging and effective, providing a solid foundation for anyone looking to improve their math skills or tackle more challenging problems. They cover a wide range of topics, including linear equations, quadratic equations, and functions, all explained in a way that is accessible to learners of all levels. Their commitment to making education accessible to everyone is commendable, and their platform is an invaluable tool for students and lifelong learners alike. Exploring their content can provide additional practice and alternative explanations that might resonate with your learning style, further cementing your grasp on these important mathematical concepts. It’s a fantastic place to continue your mathematical journey and discover the joy of problem-solving.