Solving For X In The Equation 5 - 2/(x - 8) = X
Solving algebraic equations is a fundamental skill in mathematics, and this article delves into the step-by-step process of finding all values of x that satisfy the equation 5 - 2/(x - 8) = x. This equation involves rational expressions, which require careful manipulation to avoid division by zero and ensure accurate solutions. We will explore the methodologies to tackle such equations, including identifying restrictions on x, clearing denominators, and solving the resulting polynomial equation. By the end of this exploration, you will be equipped with the knowledge to confidently solve similar algebraic problems. So, let's dive in and break down each step of solving this equation, ensuring a thorough understanding of the underlying principles and techniques.
1. Understanding the Equation and Identifying Restrictions
Before we begin manipulating the equation 5 - 2/(x - 8) = x, it's crucial to understand its structure and identify any restrictions on the variable x. This step is vital because it helps us avoid potential pitfalls, such as dividing by zero, which is undefined in mathematics. The given equation is a rational equation because it involves a fraction with a variable in the denominator. Specifically, the term 2/(x - 8) has x in the denominator, which means that the value of x cannot be such that it makes the denominator equal to zero. In other words, we need to find any values of x that would make the expression (x - 8) equal to zero. Solving the equation x - 8 = 0, we find that x = 8. Therefore, x = 8 is a restriction on the domain of the equation, meaning that x cannot be 8, as it would result in division by zero, rendering the equation undefined. This restriction is critical and must be remembered throughout the solving process. We must verify that our final solutions are not equal to 8 to ensure the validity of our answers. Identifying these restrictions at the outset helps us maintain accuracy and avoid extraneous solutions. Now that we understand the restriction on x, we can proceed with the steps to solve the equation, keeping this crucial piece of information in mind.
2. Clearing the Denominator
The next step in solving the equation 5 - 2/(x - 8) = x is to clear the denominator. Clearing the denominator simplifies the equation by eliminating the fraction, making it easier to work with. This is achieved by multiplying both sides of the equation by the least common denominator (LCD). In this case, the denominator is (x - 8), so the LCD is also (x - 8). Multiplying both sides of the equation by (x - 8), we get:
(x - 8) * [5 - 2/(x - 8)] = x * (x - 8)
We need to distribute (x - 8) on the left side of the equation. When we multiply (x - 8) by 5, we get 5(x - 8). When we multiply (x - 8) by -2/(x - 8), the (x - 8) terms cancel out, leaving us with -2. So, the left side of the equation becomes:
5(x - 8) - 2
On the right side of the equation, we multiply x by (x - 8), which gives us x(x - 8). Thus, the equation now looks like this:
5(x - 8) - 2 = x(x - 8)
This new equation no longer contains any fractions, making it a polynomial equation that is easier to solve. We will proceed to simplify and solve this equation in the next steps. By clearing the denominator, we have transformed the original rational equation into a more manageable form, setting the stage for further algebraic manipulation and ultimately, finding the solutions for x. This step is a crucial part of solving rational equations and is a common technique in algebra.
3. Expanding and Simplifying the Equation
After clearing the denominator in the equation 5 - 2/(x - 8) = x, we arrived at the simplified form 5(x - 8) - 2 = x(x - 8). The next crucial step is to expand and simplify both sides of this equation. Expanding the equation involves distributing the terms to remove the parentheses, which will help us combine like terms and rearrange the equation into a standard form. On the left side of the equation, we need to distribute the 5 across (x - 8). This means multiplying 5 by both x and -8. So, 5(x - 8) becomes 5x - 40. The left side of the equation, including the -2, is now 5x - 40 - 2. On the right side of the equation, we need to distribute the x across (x - 8). Multiplying x by both x and -8 gives us x^2 - 8x. So, the right side of the equation is x^2 - 8x. Now, the equation looks like this:
5x - 40 - 2 = x^2 - 8x
Next, we simplify the left side by combining like terms. The constants -40 and -2 can be combined to give us -42. So, the left side of the equation becomes 5x - 42. The equation now looks like this:
5x - 42 = x^2 - 8x
This simplified equation is a quadratic equation, which we can solve by setting it equal to zero and using methods such as factoring, completing the square, or the quadratic formula. By expanding and simplifying the equation, we have transformed it into a standard form that is much easier to work with. This step is essential in solving algebraic equations, as it organizes the terms and allows us to apply standard solution techniques.
4. Rearranging into a Quadratic Equation
Having expanded and simplified the equation to 5x - 42 = x^2 - 8x, the next vital step is to rearrange it into the standard form of a quadratic equation. A quadratic equation is generally expressed in the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. To rearrange our equation into this form, we need to move all terms to one side of the equation, leaving zero on the other side. In our case, it's convenient to move the terms on the left side to the right side since the x^2 term is already positive on the right side. This avoids the need to multiply through by -1, which can sometimes lead to errors. To move the terms, we subtract 5x from both sides of the equation, and we add 42 to both sides. This gives us:
5x - 42 - 5x + 42 = x^2 - 8x - 5x + 42
Simplifying both sides, we get:
0 = x^2 - 13x + 42
Now, the equation is in the standard quadratic form:
x^2 - 13x + 42 = 0
Here, we can identify that a = 1, b = -13, and c = 42. This standard form allows us to apply various methods to solve for x, such as factoring, completing the square, or using the quadratic formula. Rearranging the equation into quadratic form is a crucial step because it sets the stage for applying these standard solution techniques. Without this step, it would be much more difficult to find the values of x that satisfy the equation. This process ensures that we can utilize well-established methods to solve the quadratic equation efficiently and accurately.
5. Solving the Quadratic Equation
Now that we have rearranged the equation into the standard quadratic form x^2 - 13x + 42 = 0, we can proceed to solve for the values of x. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, factoring is a straightforward method to use if we can find two numbers that multiply to 42 and add up to -13. We look for factors of 42 and consider their sums. The pairs of factors of 42 are (1, 42), (2, 21), (3, 14), and (6, 7). Since we need the factors to add up to -13, we consider the negative pairs. The pair -6 and -7 satisfy these conditions because (-6) * (-7) = 42 and (-6) + (-7) = -13. Therefore, we can factor the quadratic equation as:
(x - 6)(x - 7) = 0
To solve for x, we set each factor equal to zero:
x - 6 = 0 or x - 7 = 0
Solving these linear equations gives us:
x = 6 or x = 7
So, the solutions to the quadratic equation are x = 6 and x = 7. However, it's crucial to remember the restriction we identified at the beginning of the process: x cannot be 8. Since neither of our solutions is 8, they are both valid solutions to the original equation. Factoring is an efficient method for solving quadratic equations when the factors are relatively easy to identify. By factoring the quadratic equation, we were able to quickly find the values of x that satisfy the equation. This step demonstrates the importance of understanding various methods for solving quadratic equations to choose the most efficient approach for a given problem.
6. Verifying the Solutions
After solving the quadratic equation x^2 - 13x + 42 = 0 and finding the solutions x = 6 and x = 7, it is essential to verify these solutions in the original equation 5 - 2/(x - 8) = x. Verification ensures that our solutions are correct and that no errors were made during the solving process. It also helps us identify any extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. First, let's verify x = 6. Substitute x = 6 into the original equation:
5 - 2/(6 - 8) = 6
Simplify the denominator:
5 - 2/(-2) = 6
Simplify the fraction:
5 - (-1) = 6
5 + 1 = 6
6 = 6
Since the equation holds true, x = 6 is a valid solution. Now, let's verify x = 7. Substitute x = 7 into the original equation:
5 - 2/(7 - 8) = 7
Simplify the denominator:
5 - 2/(-1) = 7
Simplify the fraction:
5 - (-2) = 7
5 + 2 = 7
7 = 7
Since the equation holds true, x = 7 is also a valid solution. Both solutions, x = 6 and x = 7, satisfy the original equation. Additionally, we recall that our restriction was x ≠8, and neither of our solutions violates this restriction. Therefore, we can confidently conclude that both solutions are correct. Verifying the solutions is a critical step in the problem-solving process, as it confirms the accuracy of our work and prevents errors. This step reinforces the importance of careful and thoroughness in mathematical problem-solving, ensuring that the final answers are indeed correct.
7. Conclusion: The Values of x That Satisfy the Equation
In conclusion, after meticulously solving the equation 5 - 2/(x - 8) = x, we have successfully identified all values of x that satisfy the equation. Our step-by-step approach involved understanding the equation, identifying restrictions, clearing the denominator, expanding and simplifying, rearranging into a quadratic equation, solving the quadratic equation, and, most importantly, verifying the solutions. We found that the solutions to the equation are x = 6 and x = 7. These values satisfy the original equation, and neither violates the restriction x ≠8 that we identified at the outset. Throughout this process, we employed various algebraic techniques, including clearing denominators, expanding expressions, factoring quadratic equations, and verifying solutions. Each step was crucial in ensuring the accuracy and validity of our results. This exercise demonstrates the importance of a systematic approach to solving algebraic equations, particularly those involving rational expressions. By breaking down the problem into manageable steps and applying the appropriate techniques, we were able to arrive at the correct solutions with confidence. The skills and methodologies used in this process are applicable to a wide range of mathematical problems, making this a valuable learning experience. We have not only found the solutions but also reinforced our understanding of the principles underlying algebraic problem-solving.