Solving Equations A Step By Step Guide To 4/(x-6) - 6/(x-6) = 2
Before diving into the specifics of solving the equation 4/(x-6) - 6/(x-6) = 2, it's crucial to grasp the fundamental principles that govern equation solving. At its core, solving an equation involves isolating the unknown variable, in this case, x, on one side of the equation. This isolation is achieved by performing the same operations on both sides of the equation, thereby maintaining the balance and ensuring the equality remains valid. Remember, an equation is like a balanced scale; whatever you do to one side, you must do to the other to preserve the equilibrium.
In the realm of algebraic manipulations, several key operations come into play. These include addition, subtraction, multiplication, and division. The selection of the appropriate operation hinges on the specific structure of the equation and the goal of isolating the variable. For instance, if a term is being added to the variable, subtraction is the natural antidote. Conversely, if the variable is being multiplied by a constant, division is the tool of choice. The order in which these operations are applied is also paramount, often dictated by the order of operations (PEMDAS/BODMAS), which prioritizes parentheses/brackets, exponents/orders, multiplication and division (from left to right), and addition and subtraction (from left to right).
Moreover, when dealing with equations involving fractions, as in our case, the concept of a common denominator becomes indispensable. A common denominator allows us to combine fractions, simplifying the equation and paving the way for the isolation of the variable. In the equation 4/(x-6) - 6/(x-6) = 2, the denominators are already identical, which simplifies the initial steps of the solution. However, in more complex scenarios, finding the least common multiple (LCM) of the denominators might be necessary to establish a common denominator. Once a common denominator is secured, the numerators can be combined, and the equation can be further manipulated to solve for the unknown.
Now, let's embark on a detailed, step-by-step journey to solve the equation 4/(x-6) - 6/(x-6) = 2. Our primary objective is to isolate x and unveil its value(s) that satisfy the equation. This process involves a series of algebraic manipulations, each designed to bring us closer to our goal. Remember, the key to success lies in meticulously applying each step while adhering to the fundamental principles of equation solving. Let's begin!
Step 1: Combine the Fractions
Observe that the fractions on the left-hand side of the equation share a common denominator, namely (x-6). This allows us to combine the fractions directly. We achieve this by subtracting the numerators while retaining the common denominator. The equation transforms as follows:
(4 - 6) / (x - 6) = 2
Simplifying the numerator, we get:
-2 / (x - 6) = 2
This simplification sets the stage for the next step, where we will eliminate the fraction altogether.
Step 2: Eliminate the Fraction
To rid ourselves of the fraction, we multiply both sides of the equation by the denominator, (x - 6). This operation cancels out the denominator on the left-hand side, leaving us with a simpler equation to grapple with. The transformation is as follows:
-2 / (x - 6) * (x - 6) = 2 * (x - 6)
This simplifies to:
-2 = 2(x - 6)
We have successfully eliminated the fraction, and now we are faced with a linear equation that is more amenable to solving.
Step 3: Distribute on the Right-Hand Side
To further simplify the equation, we distribute the 2 on the right-hand side across the terms within the parentheses. This involves multiplying 2 by both x and -6. The equation morphs into:
-2 = 2x - 12
This distribution expands the equation, making it clearer how to isolate x.
Step 4: Isolate the Term with x
Our next task is to isolate the term containing x, which is 2x. To achieve this, we add 12 to both sides of the equation. This cancels out the -12 on the right-hand side, leaving us with the x term isolated. The equation undergoes the following transformation:
-2 + 12 = 2x - 12 + 12
This simplifies to:
10 = 2x
Now, we have the x term isolated, and we are just one step away from finding the value of x.
Step 5: Solve for x
Finally, to solve for x, we divide both sides of the equation by the coefficient of x, which is 2. This isolates x on the right-hand side, revealing its value. The equation transforms as follows:
10 / 2 = 2x / 2
This simplifies to:
5 = x
Therefore, the solution to the equation 4/(x-6) - 6/(x-6) = 2 is x = 5.
In the realm of equation solving, the journey doesn't conclude with simply obtaining a solution. A crucial step remains: verification. Plugging the purported solution back into the original equation acts as a litmus test, confirming whether the solution holds water or if an error has crept into the process. This step is especially vital when dealing with equations involving fractions, as extraneous solutions can sometimes arise due to restrictions on the domain.
Let's subject our solution, x = 5, to this rigorous test. We substitute x = 5 back into the original equation, 4/(x-6) - 6/(x-6) = 2, and evaluate both sides independently. If the left-hand side equals the right-hand side, our solution stands vindicated. If not, it signals the need to revisit our steps and pinpoint any potential errors.
Substituting x = 5, the left-hand side becomes:
4/(5-6) - 6/(5-6) = 4/(-1) - 6/(-1) = -4 - (-6) = -4 + 6 = 2
The right-hand side of the equation is simply 2. Since the left-hand side (2) equals the right-hand side (2), our solution, x = 5, successfully navigates this test. It is indeed a valid solution to the equation.
Extraneous solutions are like imposters in the world of equation solving. They emerge as solutions during the algebraic manipulation process but fail to satisfy the original equation. This phenomenon often arises when dealing with equations involving fractions, radicals, or logarithms, where certain values might lead to undefined expressions or violate the domain of the functions involved.
In the context of equations with fractions, extraneous solutions typically stem from values that make the denominator zero. Division by zero is anathema in mathematics, leading to undefined expressions. Therefore, any value of x that causes a denominator to vanish is automatically disqualified as a valid solution. Such values might masquerade as solutions during the solving process, but they must be unmasked and discarded during verification.
In our equation, 4/(x-6) - 6/(x-6) = 2, the denominator is (x-6). Setting this denominator to zero, we get x - 6 = 0, which implies x = 6. This value, x = 6, is a potential source of trouble. If it were to emerge as a solution during our solving process, we would need to reject it as extraneous because it would lead to division by zero in the original equation.
Fortunately, our solution, x = 5, is not an extraneous one. It does not make the denominator zero, and it successfully satisfies the original equation, as we verified earlier. However, the awareness of extraneous solutions is crucial, especially when tackling more complex equations with multiple fractions or other potential pitfalls.
In this comprehensive guide, we have successfully navigated the process of solving the equation 4/(x-6) - 6/(x-6) = 2. We began by laying the groundwork, understanding the fundamental principles of equation solving and the importance of maintaining balance through algebraic manipulations. We then embarked on a step-by-step journey, combining fractions, eliminating denominators, distributing terms, and isolating x until we arrived at the solution, x = 5.
Crucially, we didn't stop there. We subjected our solution to the litmus test of verification, plugging it back into the original equation to ensure its validity. This step is paramount, especially when dealing with equations involving fractions, where extraneous solutions can lurk. We also delved into the concept of extraneous solutions, understanding their origin and the importance of identifying and discarding them.
The journey of solving equations is not merely about finding numerical answers; it's about cultivating a deep understanding of mathematical principles and developing problem-solving skills that extend far beyond the realm of algebra. Each equation is a puzzle, and the process of solving it is a rewarding exercise in logical thinking and analytical reasoning. So, embrace the challenge, and may your equation-solving endeavors be filled with success and enlightenment.
