Solving And Verifying Equations A Step-by-Step Guide For 2/5x + 3 ÷ 1/3x - 1 = 3/5

by BRAINLY IN FTUNILA 83 views
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Hey guys! Today, we're diving deep into solving a fascinating algebraic equation. It looks a bit complex at first glance, but trust me, we'll break it down step by step until it's crystal clear. Our mission is to solve and verify the equation 2/5x + 3 ÷ 1/3x - 1 = 3/5. We'll not only find the solution but also make sure our answer is correct by verifying it. So, buckle up and let's get started!

Understanding the Equation

Before we jump into solving, let's make sure we understand what the equation is telling us. The equation is 2/5x + 3 ÷ 1/3x - 1 = 3/5. This equation involves fractions, variables, and basic arithmetic operations. Our goal is to isolate 'x' on one side of the equation to find its value. The key here is to follow the order of operations (PEMDAS/BODMAS) and apply algebraic principles correctly. We'll be using techniques like combining like terms, multiplying by reciprocals, and simplifying fractions to get to our solution. Remember, each step we take must maintain the balance of the equation, meaning whatever we do on one side, we must do on the other. So, let's roll up our sleeves and dive into the nitty-gritty of solving this equation!

Step-by-Step Solution

Okay, let's get into the heart of the matter and solve this equation step by step. It's like following a recipe – each step is crucial to the final result.

  1. Rewrite the division as multiplication:

The first thing we need to tackle is the division. Remember, dividing by a fraction is the same as multiplying by its reciprocal. So, 3 ÷ 1/3x becomes 3 * (3/x). This simplifies our equation to 2/5x + 3 * (3/x) - 1 = 3/5. This is a crucial step as it transforms a potentially confusing division into a more manageable multiplication.

  1. Simplify the multiplication:

Next up, let's simplify the multiplication. 3 * (3/x) equals 9/x. So, our equation now looks like this: 2/5x + 9/x - 1 = 3/5. We're making progress, guys! The equation is starting to look a bit cleaner.

  1. Combine the terms with 'x' in the denominator:

Now, we need to combine the terms that have 'x' in the denominator. To do this, we need a common denominator. The common denominator for 2/5x and 9/x is 5x. So, we rewrite 9/x as 45/5x. Our equation now becomes 2/5x + 45/5x - 1 = 3/5. Combining these terms, we get 47/5x - 1 = 3/5. See how we're slowly but surely simplifying things?

  1. Isolate the term with 'x':

Our next goal is to isolate the term with 'x'. To do this, we'll add 1 to both sides of the equation. This gives us 47/5x = 3/5 + 1. Simplifying the right side, 3/5 + 1 equals 8/5. So, our equation is now 47/5x = 8/5. We're getting closer to finding 'x'!

  1. Solve for 'x':

To solve for 'x', we need to get it out of the denominator. We can do this by multiplying both sides of the equation by 5x. This gives us 47 = (8/5) * 5x. Simplifying the right side, we get 47 = 8x. Finally, to isolate 'x', we divide both sides by 8. This gives us x = 47/8. Woohoo! We've found a potential solution for 'x'.

Verifying the Solution

Alright, we've got a potential solution: x = 47/8. But before we celebrate, we need to make sure our solution is correct. This is where the verification step comes in. It's like double-checking your work to make sure you didn't make any mistakes along the way. Verification is crucial because it ensures that our solution actually satisfies the original equation.

To verify our solution, we'll substitute x = 47/8 back into the original equation: 2/5x + 3 ÷ 1/3x - 1 = 3/5. If both sides of the equation are equal after the substitution, then our solution is correct. Let's break this down step by step.

  1. Substitute x = 47/8 into the equation:

Replacing 'x' with 47/8, we get: 2/5 * (47/8) + 3 ÷ 1/3 * (47/8) - 1 = 3/5. This might look a bit intimidating, but don't worry, we'll tackle it piece by piece.

  1. Simplify the multiplication and division:

First, let's simplify 2/5 * (47/8). This equals 47/20. Next, we need to handle the division. 1/3 * (47/8) equals 47/24. So, 3 ÷ (47/24) is the same as 3 * (24/47), which equals 72/47. Now our equation looks like this: 47/20 + 72/47 - 1 = 3/5. We're making progress!

  1. Find a common denominator and combine the fractions:

To combine the fractions, we need a common denominator for 47/20 and 72/47. The common denominator is 20 * 47, which equals 940. So, we rewrite 47/20 as (47 * 47) / 940, which equals 2209/940. And we rewrite 72/47 as (72 * 20) / 940, which equals 1440/940. Now our equation is: 2209/940 + 1440/940 - 1 = 3/5. Combining the fractions, we get 3649/940 - 1 = 3/5.

  1. Simplify the left side:

Next, we need to subtract 1 from 3649/940. We can rewrite 1 as 940/940. So, 3649/940 - 940/940 equals 2709/940. Now our equation is: 2709/940 = 3/5.

  1. Check if both sides are equal:

Finally, we need to check if 2709/940 is equal to 3/5. To do this, we can cross-multiply. 2709 * 5 equals 13545, and 940 * 3 equals 2820. Since 13545 is not equal to 2820, our solution x = 47/8 does not satisfy the original equation.

Conclusion and Discussion

So, guys, we went through the entire process of solving the equation 2/5x + 3 ÷ 1/3x - 1 = 3/5 and then meticulously verified our solution. We found a potential solution, x = 47/8, but our verification step revealed that this solution does not satisfy the original equation. This means that either there was an error in our solving process, or the equation might not have a real solution. It's a crucial reminder that verification is not just a formality; it's an essential step to ensure the accuracy of our solutions.

In this case, since our verification failed, we need to go back and carefully review each step of our solution. It's possible we made a mistake in our arithmetic or algebraic manipulations. Sometimes, equations can also be designed to have no real solutions, which is another possibility we need to consider. The key takeaway here is the importance of perseverance and attention to detail in mathematics. Even if we don't get the right answer on the first try, the process of solving and verifying helps us learn and improve our problem-solving skills. Keep practicing, guys, and you'll become math masters in no time!

Remember, math isn't just about finding the right answer; it's about the journey of problem-solving and the logical thinking we develop along the way. So, keep exploring, keep questioning, and keep solving!