Solving 5c - 2 = 3c And Finding The Value Of 24c
Introduction
In this detailed exploration, we will delve into the realm of algebraic equations, specifically focusing on how to solve for the value of 24c when given the equation 5c - 2 = 3c. This type of problem is fundamental in algebra and often appears in various mathematical contexts, from basic equation solving to more complex applications in calculus and physics. Understanding how to manipulate and solve such equations is crucial for anyone looking to build a strong foundation in mathematics. Our discussion will begin with a step-by-step breakdown of how to isolate the variable c in the initial equation. We will then use the solved value of c to calculate the value of 24c. This process not only provides a solution to the specific problem but also illustrates the broader principles of algebraic manipulation. By working through this example, readers will gain a better understanding of how to approach similar problems and develop their algebraic problem-solving skills. Let's embark on this mathematical journey to uncover the solution together.
Understanding the Initial Equation: 5c - 2 = 3c
Our journey begins with a thorough examination of the given equation: 5c - 2 = 3c. This equation is a linear equation, a type of equation where the highest power of the variable (in this case, c) is 1. Linear equations are among the simplest yet most important types of equations in algebra. They appear frequently in various mathematical models and real-world applications. The goal in solving this equation is to isolate the variable c on one side of the equation. This means we want to rearrange the equation so that it reads c = some value. To achieve this, we will employ the fundamental principles of algebraic manipulation, which include adding, subtracting, multiplying, or dividing both sides of the equation by the same value to maintain equality. A key concept here is the preservation of balance; whatever operation is performed on one side must also be performed on the other to ensure the equation remains valid. This step-by-step process not only leads us to the solution but also reinforces the importance of maintaining mathematical integrity throughout the solving process. Understanding these principles is essential for tackling more complex algebraic problems in the future. Let's proceed with the first step in isolating c.
Step 1: Isolating the Variable 'c'
The primary goal in solving any algebraic equation is to isolate the variable. In our equation, 5c - 2 = 3c, the variable we need to isolate is c. To do this, we first need to gather all terms containing c on one side of the equation. A common strategy is to subtract the term with the smaller coefficient of c from both sides. In this case, we have 5c on the left side and 3c on the right side. Subtracting 3c from both sides will help us consolidate the c terms. This operation ensures that we maintain the balance of the equation, as whatever we do on one side, we must also do on the other. By subtracting 3c from both sides, we effectively reduce the complexity of the equation and move closer to isolating c. This step is crucial in simplifying the equation and setting the stage for further manipulation. Let's execute this step and see how it transforms the equation.
By subtracting 3c from both sides of the equation 5c - 2 = 3c, we get:
5c - 2 - 3c = 3c - 3c
This simplifies to:
2c - 2 = 0
This step is a critical simplification, bringing us closer to isolating c. Now, we have a simpler equation to work with. The next step involves further isolating the term with c by dealing with the constant term on the same side. This process of simplifying and isolating is fundamental to solving algebraic equations.
Step 2: Isolating the Constant Term
After simplifying the equation to 2c - 2 = 0, our next objective is to isolate the term containing c. Currently, we have a constant term, -2, on the same side as the c term. To eliminate this constant term, we perform the inverse operation, which in this case is adding 2 to both sides of the equation. This action maintains the balance of the equation while moving us closer to isolating c. Adding 2 to both sides effectively cancels out the -2 on the left side, leaving us with just the term containing c. This step is a crucial maneuver in the algebraic process, as it separates the variable term from the constant term, paving the way for the final isolation of c. Let's proceed with this operation and observe the transformation of the equation.
Adding 2 to both sides of the equation 2c - 2 = 0, we get:
2c - 2 + 2 = 0 + 2
This simplifies to:
2c = 2
Now, we have a much simpler equation where the c term is almost isolated. The next logical step is to eliminate the coefficient of c, which will finally give us the value of c.
Step 3: Solving for 'c'
With the equation simplified to 2c = 2, the final step in isolating c is to eliminate its coefficient, which is 2. To do this, we perform the inverse operation of multiplication, which is division. We divide both sides of the equation by 2. This action maintains the equation's balance while isolating c. Dividing both sides by 2 effectively cancels out the 2 on the left side, leaving c by itself. This is the culmination of our efforts to isolate c, and it provides us with the value of c that satisfies the original equation. This step is the final piece of the puzzle, and it reveals the solution to the first part of our problem. Let's execute this division and find the value of c.
Dividing both sides of the equation 2c = 2 by 2, we get:
(2c) / 2 = 2 / 2
This simplifies to:
c = 1
Thus, we have successfully solved for c, and we find that c = 1. Now that we have the value of c, we can move on to the second part of the problem, which is to find the value of 24c.
Calculating 24c
Now that we have determined the value of c to be 1, the next step is to calculate the value of 24c. This is a straightforward process that involves substituting the value of c into the expression 24c. The expression 24c means 24 multiplied by c. Therefore, to find the value of 24c, we simply multiply 24 by the value we found for c, which is 1. This calculation is a direct application of the value we derived for c and demonstrates how solving for a variable can be used to find the value of other expressions involving that variable. This step is crucial in completing the problem and providing the final answer. Let's perform this multiplication and find the value of 24c.
To find the value of 24c, we substitute c = 1 into the expression:
24c = 24 * 1
This simplifies to:
24c = 24
Therefore, the value of 24c is 24. This concludes the solution to the problem, where we first solved for c and then used that value to calculate 24c.
Conclusion
In summary, we have successfully navigated through the problem of finding the value of 24c given the equation 5c - 2 = 3c. Our journey began with a clear understanding of the initial equation, followed by a systematic approach to isolate the variable c. We achieved this by performing a series of algebraic manipulations, including subtracting terms from both sides and dividing to isolate c. This step-by-step process not only led us to the solution c = 1 but also reinforced the fundamental principles of algebraic equation solving. Subsequently, we utilized the value of c to calculate 24c, which we found to be 24. This final calculation underscored the practical application of solving for a variable in determining the value of other expressions. The entire process exemplifies the importance of methodical problem-solving in mathematics, where each step builds upon the previous one to reach the final answer. Understanding these principles is crucial for tackling more complex mathematical problems and developing a solid foundation in algebra. This comprehensive guide not only provides the solution but also aims to enhance the reader's understanding of the underlying mathematical concepts.
