Isolating C A Step-by-Step Guide To Solving D=(3+2c)/(c-7)

by Scholario Team 59 views

In the realm of mathematical manipulations, rearranging equations to isolate a specific variable is a fundamental skill. This article delves into the process of making cc the subject of the equation d=3+2cc−7d=\frac{3+2c}{c-7}. We'll embark on a step-by-step journey, demystifying each transformation and illuminating the underlying principles. Understanding how to manipulate equations is crucial not only in mathematics but also in various scientific and engineering disciplines where formulas need to be rearranged to solve for unknown quantities. This process involves applying algebraic operations in a strategic manner to isolate the desired variable on one side of the equation, making it the subject. As we navigate through the intricacies of this equation, we will reinforce essential algebraic concepts and problem-solving techniques. Let's begin this mathematical exploration to master the art of variable isolation.

Step 1: Clearing the Fraction

Our primary objective is to isolate cc, but it's currently entangled within a fraction. To liberate cc, we must first eliminate the denominator. The golden rule of equation manipulation dictates that whatever operation we perform on one side, we must replicate on the other. In this case, we'll multiply both sides of the equation by (c−7)(c-7):

d×(c−7)=3+2cc−7×(c−7)d \times (c-7) = \frac{3+2c}{c-7} \times (c-7)

This multiplication elegantly cancels out the denominator on the right side, leaving us with:

d(c−7)=3+2cd(c-7) = 3 + 2c

This step is crucial as it transforms the equation from a fractional form to a linear one, making it much easier to work with. By clearing the fraction, we simplify the equation and bring us closer to isolating the variable cc. This technique is a cornerstone of algebraic manipulation, applicable in a wide array of mathematical problems. It is important to remember that multiplying both sides by an expression containing a variable can sometimes introduce extraneous solutions, so it's good practice to check your final answer in the original equation. However, in this particular case, the steps we are taking do not introduce any extraneous solutions as long as ce7c e 7. The multiplication ensures that the variable of interest is no longer trapped within a fraction, allowing us to proceed with further isolation steps. The next step will involve expanding the left side of the equation and rearranging terms to group the terms containing cc on one side.

Step 2: Expanding and Rearranging

Now, we distribute dd on the left side of the equation:

dc−7d=3+2cdc - 7d = 3 + 2c

Our mission is to gather all terms containing cc on one side and all constant terms on the other. To achieve this, we'll subtract 2c2c from both sides:

dc−7d−2c=3+2c−2cdc - 7d - 2c = 3 + 2c - 2c

Simplifying, we get:

dc−7d−2c=3dc - 7d - 2c = 3

Next, we add 7d7d to both sides:

dc−7d−2c+7d=3+7ddc - 7d - 2c + 7d = 3 + 7d

This yields:

dc−2c=3+7ddc - 2c = 3 + 7d

This rearrangement is a crucial step in isolating cc. By grouping the terms with cc on one side and the constant terms on the other, we set the stage for factoring out cc. This process exemplifies the strategic manipulation of equations to achieve a desired form. The goal is to create an equation where all instances of the variable we want to isolate are on one side, making it easier to factor and eventually solve for that variable. By performing these algebraic operations systematically, we are gradually transforming the equation into a form where cc can be easily extracted. This step is a fundamental technique in algebra and is widely used in solving various types of equations. The next step will involve factoring out cc from the left side, bringing us even closer to our goal of making cc the subject of the equation.

Step 3: Factoring Out cc

Observe that cc is a common factor on the left side of the equation. We can factor it out:

c(d−2)=3+7dc(d - 2) = 3 + 7d

This factorization is a pivotal step. By extracting cc, we've effectively separated it from the other variables and constants. This brings us closer to our ultimate goal of expressing cc as the subject of the equation. Factoring is a fundamental algebraic technique that simplifies equations by identifying and extracting common factors. In this case, factoring out cc allows us to isolate it as a single term, making it easier to solve for. This step is crucial because it transforms the equation into a form where cc is multiplied by an expression, which we can then divide by to isolate cc. Factoring is a versatile technique that is used extensively in algebra and calculus to simplify expressions and solve equations. This step is a key step in solving the equation for cc, and it sets up the final step of dividing both sides by the factor of cc. The ability to recognize and apply factoring techniques is essential for algebraic manipulation and problem-solving.

Step 4: Isolating cc

To finally make cc the subject, we divide both sides of the equation by (d−2)(d - 2):

c(d−2)d−2=3+7dd−2\frac{c(d - 2)}{d - 2} = \frac{3 + 7d}{d - 2}

This simplifies to:

c=3+7dd−2c = \frac{3 + 7d}{d - 2}

Triumph! We have successfully isolated cc. This final step encapsulates the essence of equation manipulation: performing operations that progressively isolate the desired variable. Division is the inverse operation of multiplication, and in this case, it undoes the multiplication by (d−2)(d-2), leaving cc completely alone on one side of the equation. This is the culmination of our algebraic journey, where we have systematically applied various techniques to achieve our goal. It is important to note that this solution is valid as long as de2d e 2, since division by zero is undefined. This process demonstrates the power of algebraic manipulation in transforming equations and solving for specific variables. The result expresses cc in terms of dd, effectively making cc the subject of the equation. This final step is the key to solving for cc given a value for dd, and it showcases the importance of algebraic techniques in solving real-world problems. The ability to isolate variables is a fundamental skill in mathematics and science, and this example provides a clear illustration of how to accomplish this task.

Conclusion

In this comprehensive exploration, we've successfully transformed the equation d=3+2cc−7d=\frac{3+2c}{c-7} to make cc the subject, arriving at the solution c=3+7dd−2c = \frac{3 + 7d}{d - 2}. We navigated through the steps of clearing the fraction, expanding and rearranging terms, factoring out cc, and finally, isolating cc through division. This process underscores the elegance and power of algebraic manipulation. This journey highlights the importance of understanding the underlying principles of equation solving and the strategic application of algebraic operations. The ability to manipulate equations is a fundamental skill in mathematics and is essential for solving problems in various fields, including physics, engineering, and economics. By mastering these techniques, one can gain a deeper understanding of mathematical relationships and develop the problem-solving skills necessary to tackle complex problems. The process of making a variable the subject of an equation is not just a mechanical exercise; it is a testament to the power of logical reasoning and the beauty of mathematical transformations. The journey from the original equation to the final solution demonstrates the interconnectedness of mathematical concepts and the importance of a systematic approach to problem-solving.