Solving For X Equation 5x-5 Over X-2 Equals 1/3
In this article, we will delve into the step-by-step process of solving for x in the equation (5x - 5) / (x - 2) = 1/3. This type of equation, involving fractions with variables, is a common sight in algebra. Mastering the technique to solve such equations is fundamental for more advanced mathematical concepts. We will break down each step, ensuring clarity and understanding for anyone looking to enhance their algebraic skills. So, let's embark on this mathematical journey and unravel the value of x. This equation is a classic example of a rational equation, which involves algebraic fractions. Solving these equations requires careful manipulation to isolate the variable, in this case, x. The key steps typically involve eliminating the fractions, simplifying the resulting equation, and then solving for the variable. Along the way, it's crucial to be mindful of potential extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. These can arise when we multiply both sides of an equation by an expression that could be zero. Therefore, after finding potential solutions, we always check them in the original equation to ensure they are valid. In this particular equation, we'll start by cross-multiplying to get rid of the fractions. This will lead us to a linear equation, which is much easier to solve. We will then simplify the equation by combining like terms and isolate x on one side. Finally, we'll divide by the coefficient of x to find the solution. We'll also verify our solution by substituting it back into the original equation to ensure it holds true. This comprehensive approach will not only help us solve this specific equation but also provide a solid foundation for tackling similar problems in the future. Understanding the underlying principles and techniques is crucial for success in algebra and beyond. With practice and patience, solving rational equations like this becomes second nature.
Step 1: Cross-Multiplication
The first step in solving the equation (5x - 5) / (x - 2) = 1/3 is to eliminate the fractions. We achieve this by cross-multiplying. Cross-multiplication is a method used to solve equations where two fractions are set equal to each other. It involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. This effectively gets rid of the denominators, making the equation easier to work with. In our case, we multiply (5x - 5) by 3 and 1 by (x - 2). This gives us the equation 3(5x - 5) = 1(x - 2). This step is crucial because it transforms the rational equation into a linear equation, which we can solve using standard algebraic techniques. It's important to remember that cross-multiplication is valid only when we have a proportion, that is, when two fractions are equal. This method simplifies the equation by eliminating the fractions, allowing us to proceed with solving for x. After cross-multiplying, we'll have an equation without fractions, which is much easier to manipulate. This step is a fundamental technique in solving rational equations and is widely used in algebra and other branches of mathematics. By understanding and mastering cross-multiplication, you'll be able to tackle a wide range of equations involving fractions. It's a powerful tool that simplifies the process of solving for the unknown variable. Remember, the goal is to isolate x, and cross-multiplication is the first step in that direction. Once we've eliminated the fractions, we can move on to the next steps, which involve simplifying the equation and isolating x. This process of cross-multiplication is not just a shortcut; it's a logical application of the properties of equality. When we multiply both sides of an equation by the same non-zero quantity, the equality remains valid. In this case, we're effectively multiplying both sides by 3 and by (x - 2), which are the denominators of the fractions. This eliminates the fractions and gives us a linear equation that we can solve.
Step 2: Distribute and Simplify
Following the cross-multiplication, our equation now stands as 3(5x - 5) = 1(x - 2). The next step involves distributing the constants on both sides of the equation. Distribution is a fundamental property in algebra that allows us to multiply a constant across a sum or difference within parentheses. On the left side, we distribute the 3 across (5x - 5), which means we multiply 3 by both 5x and -5. This yields 15x - 15. On the right side, we distribute the 1 across (x - 2), which simply gives us x - 2, as multiplying by 1 doesn't change the expression. Our equation now looks like this: 15x - 15 = x - 2. After distributing, the next crucial step is simplification. Simplification involves combining like terms to make the equation more manageable. In this case, we have terms with x and constant terms. Our goal is to isolate x on one side of the equation, so we need to gather all the x terms on one side and the constant terms on the other. This often involves adding or subtracting terms from both sides of the equation. Remember, any operation we perform on one side of the equation must also be performed on the other side to maintain equality. Simplifying the equation is a key step in solving for x. It reduces the complexity of the equation and makes it easier to isolate the variable. By combining like terms, we're essentially rearranging the equation into a more standard form, which we can then solve using basic algebraic techniques. This step is not just about making the equation look simpler; it's about making it mathematically easier to work with. The simplified equation will have fewer terms and will be in a form that's easier to manipulate. This process of distribution and simplification is a cornerstone of algebra. It's used in a wide variety of algebraic problems, from solving simple equations to more complex tasks like factoring and expanding expressions. Mastering these techniques is crucial for success in algebra and beyond. With practice, you'll become more comfortable with these steps and be able to apply them quickly and efficiently. Remember, the goal is to get x by itself on one side of the equation, and distribution and simplification are essential tools in achieving that goal.
Step 3: Isolate x
After simplifying the equation to 15x - 15 = x - 2, our next objective is to isolate the variable x. Isolating x means getting x by itself on one side of the equation. This is a crucial step in solving for x because once x is isolated, we can directly read off its value. To isolate x, we need to move all terms containing x to one side of the equation and all constant terms to the other side. This is typically done by adding or subtracting terms from both sides of the equation. In our case, we can start by subtracting x from both sides of the equation. This will move the x term from the right side to the left side. The equation then becomes 15x - x - 15 = -2. Simplifying the left side, we get 14x - 15 = -2. Next, we need to move the constant term, -15, to the right side of the equation. We can do this by adding 15 to both sides. This gives us 14x = -2 + 15, which simplifies to 14x = 13. Now, we have all the x terms on one side and all the constant terms on the other. The final step in isolating x is to divide both sides of the equation by the coefficient of x, which in this case is 14. This will give us the value of x. Remember, the key to isolating x is to perform the same operations on both sides of the equation to maintain equality. Each step we take brings us closer to the solution. This process of isolating the variable is a fundamental technique in algebra and is used extensively in solving equations of various types. By mastering this technique, you'll be able to solve a wide range of algebraic problems. It's important to be methodical and careful with each step to avoid errors. Check your work as you go to ensure you're on the right track. With practice, isolating the variable will become second nature, and you'll be able to solve equations with confidence.
Step 4: Solve for x
Having isolated x in the previous step, we arrived at the equation 14x = 13. Now, the final step is to solve for x. Solving for x in this context means finding the value of x that satisfies the equation. To do this, we need to get x completely by itself on one side of the equation. As we have 14x = 13, we can solve for x by dividing both sides of the equation by 14. This is because dividing by the coefficient of x will leave x by itself. So, dividing both sides by 14, we get x = 13/14. This is the solution to the equation. It means that if we substitute 13/14 for x in the original equation, the equation will hold true. The solution x = 13/14 is a fractional value, which is perfectly acceptable. Solutions to algebraic equations can be integers, fractions, or even irrational numbers. It's important to be comfortable working with all types of numbers when solving equations. After finding the solution, it's always a good practice to check your answer by substituting it back into the original equation. This helps to ensure that you haven't made any mistakes along the way. In our case, we would substitute 13/14 for x in the equation (5x - 5) / (x - 2) = 1/3 and see if the equation holds true. This step of solving for x is the culmination of all the previous steps. It's the point where we finally find the value of the unknown variable. It's a rewarding feeling to arrive at the solution after carefully working through the steps. Remember, solving for x is not just about finding a number; it's about understanding the relationships between the different parts of the equation and using algebraic techniques to unravel those relationships. With practice, you'll become more proficient at solving for x in a wide variety of equations. This is a fundamental skill in algebra and is essential for success in more advanced mathematical topics.
Step 5: Verification
After finding the solution x = 13/14, it's crucial to verify our answer. Verification involves substituting the solution back into the original equation to ensure it holds true. This step is essential because it helps us catch any errors we might have made during the solving process. It also confirms that our solution is valid and satisfies the original equation. To verify our solution, we substitute x = 13/14 into the original equation (5x - 5) / (x - 2) = 1/3. This means replacing every instance of x in the equation with 13/14. The equation then becomes (5*(13/14) - 5) / ((13/14) - 2) = 1/3. Now, we need to simplify both the numerator and the denominator of the fraction on the left side of the equation. Let's start with the numerator: 5*(13/14) - 5 = (65/14) - 5. To subtract 5 from 65/14, we need to express 5 as a fraction with a denominator of 14. So, 5 = 70/14. Therefore, (65/14) - (70/14) = -5/14. Now, let's simplify the denominator: (13/14) - 2. Again, we need to express 2 as a fraction with a denominator of 14. So, 2 = 28/14. Therefore, (13/14) - (28/14) = -15/14. Our equation now looks like this: (-5/14) / (-15/14) = 1/3. To divide fractions, we multiply by the reciprocal of the divisor. So, (-5/14) / (-15/14) = (-5/14) * (-14/15). The 14s cancel out, and we're left with (-5/-15), which simplifies to 1/3. So, the left side of the equation equals 1/3, which is the same as the right side of the equation. This confirms that our solution x = 13/14 is correct. Verification is a powerful tool that helps us build confidence in our solutions. It's a step that should never be skipped, especially in algebra, where errors can easily occur. By verifying our solutions, we ensure that we're not only getting the right answer but also understanding the process and the underlying principles. This step is not just about checking our work; it's about solidifying our understanding and reinforcing our problem-solving skills. With practice, verification becomes a natural part of the problem-solving process, and you'll develop a habit of checking your answers to ensure accuracy.
In conclusion, solving for x in the equation (5x - 5) / (x - 2) = 1/3 involves a series of algebraic steps, each crucial to arriving at the correct solution. We began by cross-multiplying to eliminate the fractions, transforming the equation into a more manageable form. This step is a fundamental technique in solving rational equations and is widely used in algebra. Next, we distributed and simplified the equation, combining like terms to make it easier to isolate the variable. This process of distribution and simplification is a cornerstone of algebra and is used in a wide variety of algebraic problems. Then, we isolated x, moving all terms containing x to one side of the equation and all constant terms to the other side. This step is crucial because once x is isolated, we can directly read off its value. After isolating x, we solved for x, finding the value of x that satisfies the equation. The solution we found was x = 13/14, a fractional value that is perfectly acceptable in algebraic solutions. Finally, we verified our solution by substituting it back into the original equation. This step is essential because it helps us catch any errors we might have made during the solving process and confirms that our solution is valid. The verification process confirmed that x = 13/14 is indeed the correct solution. This comprehensive approach to solving for x not only provides the answer but also reinforces the underlying algebraic principles and techniques. By mastering these techniques, you'll be able to tackle a wide range of algebraic problems with confidence. Remember, solving for x is not just about finding a number; it's about understanding the relationships between the different parts of the equation and using algebraic techniques to unravel those relationships. With practice, you'll become more proficient at solving for x in a wide variety of equations. This is a fundamental skill in algebra and is essential for success in more advanced mathematical topics. The process we've outlined here is a systematic approach that can be applied to many similar problems. It involves careful manipulation of the equation, step-by-step simplification, and a final check to ensure accuracy. This methodical approach is key to success in algebra and beyond. By following these steps, you can confidently solve for x in a variety of equations and build a strong foundation in algebraic problem-solving.
