Solving For F: A Step-by-Step Guide

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Solving for f: A Step-by-Step Guide

Hey guys! Let's dive into solving this equation for 'f': (1/3)(6f + 5) + (2/3)(3f - 1) = (4/3)f. It might look a little intimidating at first, but don't worry, we'll break it down step by step. We're going to make it super easy, so grab your pencils, and let's get started!

1. Understanding the Equation

Before we jump into solving, let's take a good look at what we're dealing with. The equation we're tackling is: (1/3)(6f + 5) + (2/3)(3f - 1) = (4/3)f. This equation involves fractions, parentheses, and our variable, 'f'. The goal here is to isolate 'f' on one side of the equation so we can determine its value.

  • Why is this important? Well, solving equations like this is a fundamental skill in algebra. It helps us understand relationships between quantities and solve real-world problems. Imagine you're trying to figure out how much to charge for a product, or how long it will take to travel a certain distance – these are the types of scenarios where understanding equations comes in handy.
  • Breaking it Down: We have two terms on the left side of the equation that involve fractions multiplied by expressions in parentheses. On the right side, we have a single term with a fraction and 'f'. Our strategy will be to simplify both sides, get rid of the fractions, and then isolate 'f'.

Remember, mathematics is like a puzzle. Each piece needs to fit perfectly to reveal the solution. So, let's start fitting these pieces together!

2. Distributing the Fractions

The first step in simplifying our equation is to get rid of those parentheses. We do this by distributing the fractions outside the parentheses to the terms inside. This means we'll multiply (1/3) by both 6f and 5, and then multiply (2/3) by both 3f and -1. Let's see how this looks:

  • (1/3)(6f + 5):
    • (1/3) * 6f = 2f
    • (1/3) * 5 = 5/3
    • So, (1/3)(6f + 5) becomes 2f + 5/3
  • (2/3)(3f - 1):
    • (2/3) * 3f = 2f
    • (2/3) * -1 = -2/3
    • So, (2/3)(3f - 1) becomes 2f - 2/3

Now, let's rewrite our equation with these simplified terms:

2f + 5/3 + 2f - 2/3 = (4/3)f

See? We're already making progress! By distributing the fractions, we've made the equation look a little less cluttered. This step is crucial because it allows us to combine like terms, which is what we'll do next. Think of it as decluttering your workspace before starting a big project – it makes everything easier to manage.

3. Combining Like Terms

Now that we've distributed the fractions, it's time to combine the like terms on the left side of the equation. Like terms are those that have the same variable raised to the same power (in this case, 'f') or are constants (just numbers). In our equation, we have '2f' and '2f' as like terms, and '5/3' and '-2/3' as like terms.

  • Combining the 'f' terms:
    • 2f + 2f = 4f
  • Combining the constants:
    • 5/3 - 2/3 = 3/3 = 1

So, the left side of our equation simplifies to 4f + 1. Now, let's rewrite the entire equation:

4f + 1 = (4/3)f

Doesn't that look much cleaner? By combining like terms, we've reduced the complexity of the equation. This step is like organizing your ingredients before you start cooking – it makes the rest of the process much smoother. Next, we'll want to isolate the 'f' terms on one side of the equation.

4. Isolating the 'f' Terms

Our next goal is to get all the 'f' terms on one side of the equation and the constants on the other. Currently, we have '4f' on the left side and '(4/3)f' on the right side. To isolate the 'f' terms, we can subtract '(4/3)f' from both sides of the equation. This will eliminate the 'f' term on the right side and move it to the left side.

  • Subtracting (4/3)f from both sides:
    • 4f + 1 - (4/3)f = (4/3)f - (4/3)f
    • This simplifies to: 4f - (4/3)f + 1 = 0

Now, we need to subtract the fractions. To do this, we'll rewrite '4f' as a fraction with a denominator of 3:

  • 4f = (4/1)f = (12/3)f

So our equation becomes:

(12/3)f - (4/3)f + 1 = 0

Now we can subtract the 'f' terms:

  • (12/3)f - (4/3)f = (8/3)f

Our equation now looks like this:

(8/3)f + 1 = 0

We're getting closer! We've successfully moved all the 'f' terms to the left side. Now, we need to isolate the 'f' term further by moving the constant to the other side.

5. Isolating 'f'

We're on the home stretch now! We have the equation (8/3)f + 1 = 0. To isolate 'f', we need to get rid of the '+ 1' on the left side. We can do this by subtracting 1 from both sides of the equation:

  • (8/3)f + 1 - 1 = 0 - 1
  • This simplifies to: (8/3)f = -1

Now, 'f' is almost completely isolated. We just need to get rid of the fraction (8/3) that's multiplying 'f'. To do this, we can multiply both sides of the equation by the reciprocal of (8/3), which is (3/8):

  • (3/8) * (8/3)f = -1 * (3/8)
  • On the left side, (3/8) * (8/3) cancels out, leaving us with just 'f':
  • f = -1 * (3/8)
  • Multiplying -1 by (3/8) gives us:
  • f = -3/8

And there we have it! We've successfully solved for 'f'.

6. The Solution

After all that work, we've found that f = -3/8. This is the value of 'f' that makes the original equation true. Awesome job, guys! You've tackled a multi-step equation and come out on top.

  • Let's recap what we did:
    1. We distributed the fractions to eliminate the parentheses.
    2. We combined like terms to simplify the equation.
    3. We isolated the 'f' terms on one side of the equation.
    4. We isolated 'f' completely by multiplying by the reciprocal.

Each of these steps is a fundamental technique in algebra, and mastering them will help you solve a wide range of equations.

7. Why This Matters

Okay, so we solved for 'f'. But why does this matter in the real world? Well, algebraic equations are used everywhere, from engineering and physics to economics and computer science. Understanding how to solve them is crucial for anyone pursuing a career in these fields.

  • Real-World Examples:
    • Engineering: Engineers use equations to design bridges, buildings, and machines.
    • Physics: Physicists use equations to describe the motion of objects and the behavior of energy.
    • Economics: Economists use equations to model economic systems and predict market trends.
    • Computer Science: Computer scientists use equations to develop algorithms and software.

Even in everyday life, we use math concepts without even realizing it. For example, when you're calculating a tip at a restaurant or figuring out how much paint you need for a room, you're using basic algebraic principles.

Conclusion

So, there you have it! We've successfully solved for 'f' in the equation (1/3)(6f + 5) + (2/3)(3f - 1) = (4/3)f. Remember, the key to solving complex problems is to break them down into smaller, more manageable steps. By distributing, combining like terms, and isolating the variable, we can conquer even the trickiest equations. Keep practicing, guys, and you'll become math whizzes in no time!