Solve Algebraic Equation: (1/5)f + (2/3) - 2 = F - (2/5)

by Andrew McMorgan 57 views

Hey mathletes! Welcome back to Plastik Magazine, where we tackle those brain-bending problems that make you go "huh?". Today, we're diving deep into the world of algebra to solve the equation (1/5)f + (2/3) - 2 = f - (2/5). This might look a bit intimidating with all those fractions, guys, but trust me, by the end of this, you'll be a pro at handling these types of problems. We're going to break it down step-by-step, making sure every move we make is clear and logical. So, grab your virtual calculators, settle in, and let's get this equation sorted!

Understanding the Equation and Our Goal

So, what are we actually trying to do here? When we talk about solving an equation like (1/5)f + (2/3) - 2 = f - (2/5), our main mission is to find the value of the unknown variable, which in this case is 'f'. Think of 'f' as a mystery number that makes both sides of the equation perfectly equal. Our job is to isolate 'f' on one side of the equals sign. To do this, we'll use a set of algebraic rules that are like the golden rules of math: whatever you do to one side of the equation, you must do to the other. This keeps the balance true! We're going to work through this equation by performing operations like adding, subtracting, multiplying, and dividing. The key is to move all the terms containing 'f' to one side and all the constant numbers to the other. Dealing with fractions can be a bit tricky, so we'll also touch upon strategies to make those parts easier to handle. The ultimate aim is to get an answer in the form of 'f = [some number]'. This number will be the solution that satisfies the original equation. We'll also double-check our answer at the end, which is a super important step in algebra to make sure we haven't made any slip-ups along the way. So, let's get ready to unpack this equation and reveal the value of 'f'!

Step 1: Simplify Both Sides (If Possible)

Alright team, the first strategic move when you're faced with an equation like (1/5)f + (2/3) - 2 = f - (2/5) is to see if you can simplify either side. In this particular equation, the left side has a constant term (-2) that can be combined with the fraction (2/3). Combining these constants will make the equation a little cleaner to work with. To combine 2/3 and -2, we need a common denominator. The number 2 can be written as 2/1. The common denominator between 3 and 1 is 3. So, we'll rewrite 2 as 6/3. Now, we can combine 2/3 and -6/3, which gives us (2 - 6)/3 = -4/3. So, the left side of our equation becomes (1/5)f - 4/3. The right side, f - (2/5), is already as simple as it can get. This initial simplification step is crucial because it reduces the number of terms we have to juggle, making subsequent steps less prone to errors. It's like clearing the clutter before you start building something important. By simplifying, we're making the path to isolating 'f' much clearer. It shows that even complex-looking equations can often be broken down into more manageable parts. So, remember, always look for opportunities to simplify first. This is a fundamental principle in algebra that saves a lot of headache down the line. It's all about making the problem work for you, not the other way around. This proactive approach is what separates those who struggle with math from those who conquer it. Keep this in mind for all your future algebraic adventures!

Step 2: Eliminate Fractions by Finding a Common Denominator

Now that we've simplified the left side, our equation looks like this: (1/5)f - 4/3 = f - (2/5). The next big hurdle, or as I like to call it, the 'fraction fiesta', is dealing with these denominators: 5, 3, and 5 again. To get rid of these pesky fractions and make our lives easier, we're going to find the least common multiple (LCM) of all the denominators. The denominators here are 5 and 3. The multiples of 5 are 5, 10, 15, 20... and the multiples of 3 are 3, 6, 9, 12, 15, 18... See that? 15 is the smallest number that appears in both lists. So, our Least Common Denominator (LCD) is 15. Now, here's the magic trick: we're going to multiply every single term in the equation by this LCD, which is 15. This is totally allowed because multiplying both sides of an equation by the same non-zero number keeps it balanced. Let's do it:

15 * [(1/5)f] - 15 * [4/3] = 15 * [f] - 15 * [2/5]

Let's crunch these numbers:

  • 15 * (1/5)f = (15/5)f = 3f (The 15 and 5 cancel out, leaving 3)
  • 15 * (4/3) = (15 * 4) / 3 = 60 / 3 = 20 (The 15 and 3 cancel, leaving 5 * 4)
  • 15 * f = 15f
  • 15 * (2/5) = (15 * 2) / 5 = 30 / 5 = 6 (The 15 and 5 cancel, leaving 3 * 2)

So, after multiplying every term by 15, our equation transforms from (1/5)f - 4/3 = f - (2/5) into 3f - 20 = 15f - 6. Look at that! All the fractions are gone. This step is super powerful because it converts a fractional equation into a much simpler linear equation that we can solve using basic operations. It's a game-changer, guys, and a technique you'll use constantly when dealing with fractions in algebra. It really streamlines the process and makes the next steps a breeze. Remember, finding that LCD and multiplying through is your secret weapon against those fractions!

Step 3: Group 'f' Terms on One Side

We've successfully banished the fractions, and our equation is now 3f - 20 = 15f - 6. Our next objective is to gather all the terms that have 'f' in them onto one side of the equation and all the constant numbers onto the other. This is where we start really isolating our variable. Typically, it's easier to move the 'f' terms to the side where the coefficient of 'f' will be positive, but either way works. Let's choose to move the 'f' terms to the right side. To do this, we need to eliminate the '3f' from the left side. We can do this by subtracting '3f' from both sides of the equation:

3f - 20 - 3f = 15f - 6 - 3f

On the left side, the '+3f' and '-3f' cancel each other out, leaving us with just -20. On the right side, we combine the 'f' terms: 15f - 3f gives us 12f. So, our equation now looks like this: -20 = 12f - 6.

This is great! We've successfully moved all the 'f' terms to one side. Now, all that's left is to get the constant terms together. See how we've systematically simplified the equation? Each step builds on the last, making the problem progressively easier to solve. This methodical approach is key in algebra. It's not about being a genius; it's about being organized and following the rules. We're getting closer and closer to finding out what 'f' is worth. Keep up the awesome work, everyone!

Step 4: Group Constant Terms on the Other Side

We're on the home stretch, folks! Our equation is currently -20 = 12f - 6. We've got the 'f' terms on the right side, and now we need to move the constant term '-6' from the right side to the left side, where all the other constants are gathering. To get rid of the '-6' on the right side, we do the opposite operation: we add 6 to both sides of the equation. This keeps everything balanced, remember?

-20 + 6 = 12f - 6 + 6

Let's do the math:

  • On the left side: -20 + 6 = -14
  • On the right side: -6 + 6 = 0, so we're left with just 12f.

Our equation has now transformed into -14 = 12f. How cool is that? We've successfully isolated the term with 'f' on one side and all the constants on the other. This is the most critical part of solving for a variable. We've systematically used addition and subtraction to move terms around, and now we're just one step away from the final answer. This process demonstrates the power of inverse operations in algebra – using addition to undo subtraction, and vice versa. It's all about isolating the unknown. We're almost there, guys! Just one final step to go.

Step 5: Isolate 'f' by Division

We've arrived at the final frontier: -14 = 12f. Our goal is to find the value of 'f', and right now, 'f' is being multiplied by 12. To isolate 'f', we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the equation by 12:

-14 / 12 = 12f / 12

Let's simplify:

  • On the left side: -14 / 12. Both 14 and 12 are divisible by 2. So, -14 ÷ 2 = -7, and 12 ÷ 2 = 6. This gives us the fraction -7/6.
  • On the right side: 12f / 12. The 12s cancel out, leaving us with just f.

So, our solution is f = -7/6.

And there you have it! We've successfully solved the equation (1/5)f + (2/3) - 2 = f - (2/5). The value of 'f' that makes this equation true is -7/6. This entire process, from simplifying to isolating the variable, is a fundamental skill in algebra. It involves understanding that equations are balanced and that operations must be applied equally to both sides. Dealing with fractions is common, and as you saw, finding a common denominator and multiplying through is a great strategy to simplify the problem. Remember these steps for any similar algebraic equations you encounter. Keep practicing, and you'll become a fraction-solving wizard in no time!

Step 6: Verification (Checking Your Answer)

It's super important in math, especially in algebra, to always check your answer. This is like proofreading your work to make sure you didn't miss any typos or make any silly mistakes. We found that f = -7/6. Let's plug this value back into the original equation: (1/5)f + (2/3) - 2 = f - (2/5) and see if both sides are equal.

Left Side: (1/5) * (-7/6) + (2/3) - 2 = -7/30 + 2/3 - 2

To add these, we need a common denominator, which is 30.

= -7/30 + (210)/(310) - (2*30)/30 = -7/30 + 20/30 - 60/30 = (-7 + 20 - 60) / 30 = (13 - 60) / 30 = -47/30

Right Side: f - (2/5) = (-7/6) - (2/5)

To subtract these, we need a common denominator, which is 30.

= (-75)/(65) - (26)/(56) = -35/30 - 12/30 = (-35 - 12) / 30 = -47/30

Boom! Both sides equal -47/30. This confirms that our solution f = -7/6 is absolutely correct. This verification step is non-negotiable, guys. It gives you confidence in your answer and helps solidify your understanding of algebraic principles. Never skip it!

Conclusion

So, there you have it! We've successfully navigated the intricacies of solving the algebraic equation (1/5)f + (2/3) - 2 = f - (2/5), arriving at the solution f = -7/6. We broke down the problem into manageable steps: simplifying, eliminating fractions by finding a common denominator, grouping like terms, and finally isolating the variable. Remember the power of finding that Least Common Denominator (LCD) – it's a true lifesaver when dealing with fractional equations. And don't forget the crucial step of verifying your answer by plugging it back into the original equation. It’s your guarantee of accuracy and a fantastic way to build confidence in your algebraic skills. Keep practicing these techniques, and you'll find that tackling even more complex equations becomes second nature. Algebra is all about logical steps and consistent application of rules. So, keep that brain sharp, keep solving, and keep shining! See you in the next article, math wizards!