Solving For 'c': A Step-by-Step Guide

Hey guys! Let's dive into solving the equation (c - 10)/9 = -1/c. This is a fun problem that involves a bit of algebra, and I'll walk you through it step-by-step so you can totally nail it. We'll be using some basic algebraic principles to isolate 'c' and find its value. So, grab your pencils and let's get started! Our goal is to find the value(s) of the variable 'c' that make the equation true. This means we need to manipulate the equation until we get 'c' all by itself on one side. This process involves a few key steps: cross-multiplication, simplifying the resulting equation, and then solving the quadratic equation that we'll end up with. It's like a puzzle, and we're going to put all the pieces together. The beauty of algebra is that it gives us a systematic way to solve for unknown variables, and this problem is a perfect example of how that works. Don't worry if it seems a bit tricky at first; with practice, you'll become a pro at solving these types of equations. Ready? Let's go!

Step 1: Cross-Multiplication

Alright, the first step is to get rid of those fractions. We can do this by cross-multiplying. Basically, we're going to multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. This gives us (c - 10) * c = -1 * 9. Easy peasy, right? Cross-multiplication is a super useful technique for solving equations that involve fractions. It allows us to transform the equation into a more manageable form, eliminating the fractions and making it easier to work with. Remember, the key is to multiply the numerator of one fraction by the denominator of the other, ensuring that we account for both sides of the equation. This step is all about making the equation cleaner and preparing it for further simplification. So, we're going to multiply 'c' by 'c - 10' on the left side and '-1' by '9' on the right side. This will give us a new equation without any fractions, which will be much easier to handle. Now, we are ready to proceed with the next step, which involves expanding and simplifying the equation to eventually solve for 'c'.

Expanding and Simplifying

Now, let's expand the left side of the equation. We multiply 'c' by both terms inside the parentheses: c * c - 10 * c. This gives us c

2 - 10c. On the right side, -1 * 9 equals -9. So, our equation now looks like this: c

2 - 10c = -9. We've simplified the equation by removing the fractions and expanding the terms. This expansion is essential for revealing the underlying structure of the equation, setting the stage for further simplification and ultimately solving for 'c'. This is the foundation upon which we'll build our solution. With each step, the equation becomes clearer and easier to manage, inching us closer to our goal. Remember, attention to detail is key in algebra. Make sure you multiply each term correctly and keep track of your signs. The goal here is to get all the terms on one side of the equation and set it equal to zero. This is a standard form for solving quadratic equations, and we'll see why it's important in the next step. So, get ready to move to the next stage where we will arrange the equation in standard form. Now, the equation is getting closer to a form we can solve.

Step 2: Rearrange into Standard Quadratic Form

Next up, we want to rearrange the equation into standard quadratic form, which is ax

2 + bx + c = 0. To do this, we need to move the -9 from the right side to the left side. We do this by adding 9 to both sides of the equation. This gives us c

2 - 10c + 9 = 0. We've got our equation in the perfect form for solving using different methods. The standard quadratic form is a fundamental concept in algebra, allowing us to recognize and address these equations systematically. By rearranging our equation into this form, we're preparing it for the next steps, where we'll solve for 'c' using various methods. Make sure that all terms are on the same side and that the equation is set equal to zero. The standard form provides a clear structure, making it easier to see and apply solution methods. This step is crucial because it sets the stage for solving the quadratic equation using methods like factoring, completing the square, or the quadratic formula. Each of these methods will rely on the equation being in standard form to work correctly. This step is like setting the scene for the grand finale. Now, the equation is ready to reveal its secrets!

Methods for Solving Quadratic Equations

There are several ways to solve a quadratic equation like the one we have: Factoring, Completing the Square, and the Quadratic Formula. Factoring involves finding two numbers that multiply to give you the constant term (9 in this case) and add up to the coefficient of the 'c' term (-10). Completing the square is another technique. It involves manipulating the equation to create a perfect square trinomial on one side. The quadratic formula is a universal tool that works for any quadratic equation, regardless of whether it can be factored easily. Each of these methods has its advantages and disadvantages, and the best choice depends on the specific equation and your personal preference. For this example, let's use factoring. We'll look for two numbers that multiply to 9 and add up to -10. These numbers are -1 and -9, so we can factor the equation into (c - 1)(c - 9) = 0. Once we've factored the equation, we can set each factor equal to zero and solve for 'c'. This leads us to our final step, where we'll find the values of 'c' that make the original equation true. The choice of method depends on the equation's structure and your comfort level. Remember, practice is key! So, we will select one method in order to show you a practical example of how to solve the question.

Step 3: Solve for 'c'

Now that we've factored the equation into (c - 1)(c - 9) = 0, we can solve for 'c'. We set each factor equal to zero: c - 1 = 0 and c - 9 = 0. Solving these simple equations, we get c = 1 and c = 9. Congratulations! We've found the solutions for 'c'. These two values are the solutions to the original equation. The equation (c - 10)/9 = -1/c is true when c = 1 and when c = 9. These are the values that satisfy the original equation. Each of these values represents a point where the equation holds true, providing a complete solution to the problem. The final step is where we find the values of 'c' that make the original equation true. We've successfully used factoring to identify these values, and they represent the solutions to our problem. So, c = 1 and c = 9 are the two solutions to the equation (c - 10)/9 = -1/c. This is a perfect example of how we use the different methods to solve the question.

Checking Your Solutions

It's always a good idea to check your solutions to make sure they're correct. We can do this by plugging the values of 'c' back into the original equation. Let's start with c = 1: (1 - 10)/9 = -1/1. This simplifies to -9/9 = -1, which is true. Now, let's check c = 9: (9 - 10)/9 = -1/9. This simplifies to -1/9 = -1/9, which is also true. Both of our solutions check out, meaning we've solved the equation correctly. This step is about verifying that your answers are correct and that you haven't made any mistakes along the way. Plugging the values back into the original equation and checking them ensures that your solutions are valid. Checking is a crucial part of problem-solving. It's like double-checking your work to make sure you didn't miss anything. Always verify your solutions to ensure their validity and accuracy. It's a quick way to confirm that your answers are correct and that you've successfully solved the equation.

Conclusion

And there you have it, guys! We've successfully solved the equation (c - 10)/9 = -1/c. We found that c = 1 and c = 9 are the solutions. We started with cross-multiplication, rearranged the equation, factored it, and finally found the values of 'c'. Remember, practice makes perfect. Keep working on these types of problems, and you'll become a pro in no time. If you've got any questions or want to try another problem, just let me know. Keep up the great work! This whole process highlights the power of algebra in solving complex problems systematically. You've demonstrated how to manipulate the equation, solve, and check your work. Keep practicing, and you'll master these types of questions. Congratulations on your achievement!