Solving Equations A Step By Step Guide For (2/3)x - (1/9)x + 5 = 20

Hey guys! Today, we're going to dive into solving a simple algebraic equation. It might look a bit intimidating at first, with those fractions hanging around, but trust me, it's totally manageable. We'll break it down step by step, so you'll be solving for x like a pro in no time. Our equation is:

(2/3)x - (1/9)x + 5 = 20

So, let's get started and untangle this equation together!

Step 1: Combining Like Terms

The first thing we need to do is to simplify the equation by combining the terms that have 'x' in them. We have two terms with 'x': (2/3)x and -(1/9)x. To combine these, we need to find a common denominator for the fractions 2/3 and 1/9. The least common multiple of 3 and 9 is 9, so that's going to be our common denominator.

Let's rewrite 2/3 as an equivalent fraction with a denominator of 9. To do this, we multiply both the numerator and the denominator of 2/3 by 3: (2 * 3) / (3 * 3) = 6/9. Now we can rewrite our equation as:

(6/9)x - (1/9)x + 5 = 20

Now that the fractions have the same denominator, we can easily subtract them. Think of it like this: if you have 6 ninths of something and you take away 1 ninth, how many ninths do you have left? That's right, 5 ninths! So, we have:

(5/9)x + 5 = 20

Awesome! We've combined the 'x' terms and simplified our equation quite a bit. This step is crucial because it makes the equation much easier to work with. By combining like terms, we've reduced the number of individual elements we need to deal with, which streamlines the rest of the solving process. It’s like decluttering your workspace before starting a big project – you'll have a much clearer view of what needs to be done.

Think of it this way: imagine you're trying to count a pile of coins. If you have a mix of pennies, nickels, and dimes scattered all over the place, it's going to take you a while to get an accurate count. But if you first group all the pennies together, then all the nickels, and then all the dimes, it becomes much easier to count each group separately and then add them up. Combining like terms in an equation is like sorting those coins – it helps you organize the information and makes the next steps much smoother.

This principle of simplifying by combining like terms applies not only to this specific equation but to algebra in general. Whenever you encounter an equation with multiple terms involving the same variable or constant, your first instinct should be to look for opportunities to combine them. This skill is fundamental for solving more complex equations and tackling advanced mathematical concepts down the road.

So, remember this first step: identify the like terms, find a common denominator if necessary, and then combine them. It's a foundational technique that will serve you well throughout your mathematical journey. Now, let’s move on to the next step in solving our equation!

Step 2: Isolating the x Term

Our next goal is to isolate the term with 'x' on one side of the equation. Right now, we have (5/9)x + 5 = 20. We need to get rid of that '+ 5' that's hanging out on the left side. To do this, we use the principle of inverse operations. Since we're adding 5, we need to do the opposite, which is subtracting 5.

Here's the key thing to remember: whatever you do to one side of the equation, you must do to the other side to keep things balanced. It's like a scale – if you take something off one side, you need to take the same amount off the other side to keep it level. So, we're going to subtract 5 from both sides of the equation:

(5/9)x + 5 - 5 = 20 - 5

On the left side, the +5 and -5 cancel each other out, leaving us with just (5/9)x. On the right side, 20 - 5 equals 15. So our equation now looks like this:

(5/9)x = 15

Great job! We've successfully isolated the term with 'x' on one side of the equation. This is a really important step because it brings us one step closer to finding the value of x itself. By isolating the 'x' term, we're essentially peeling away the layers of the equation, like unwrapping a present, until we get to the core – the value of x.

The idea of isolating a variable is a cornerstone of algebra. It’s not just about solving this particular equation; it’s about developing a fundamental skill that you'll use over and over again in mathematics and even in other fields. Think of it as learning a basic move in a game – once you've mastered it, you can use it in countless situations and build upon it to learn more complex moves.

For example, in physics, you might need to isolate a variable to calculate velocity or acceleration. In chemistry, you might need to isolate a variable to determine the concentration of a solution. Even in everyday life, the ability to isolate a variable can be useful. Imagine you're trying to figure out how much you can spend on groceries this week. You might have a total budget and some fixed expenses, and you need to isolate the amount you can spend on groceries. The same principle applies – you need to manipulate the equation to get the variable you're interested in (in this case, the grocery budget) by itself on one side.

So, mastering the art of isolating variables is a skill that extends far beyond the classroom. It's a powerful tool for problem-solving and critical thinking in a wide range of contexts. Now that we've isolated the 'x' term in our equation, we're ready for the final step: solving for x itself. Let's move on and finish this problem!

Step 3: Solving for x

We're in the home stretch now! We have the equation (5/9)x = 15. Remember, (5/9)x means (5/9) * x. So, to get 'x' by itself, we need to undo the multiplication by (5/9). The inverse operation of multiplying by a fraction is multiplying by its reciprocal. The reciprocal of 5/9 is 9/5. So, we're going to multiply both sides of the equation by 9/5:

(9/5) * (5/9)x = 15 * (9/5)

On the left side, (9/5) * (5/9) cancels out to 1, leaving us with just 'x'. That's exactly what we wanted! On the right side, we have 15 * (9/5). We can think of 15 as 15/1, so we're multiplying (15/1) * (9/5). To multiply fractions, we multiply the numerators and multiply the denominators:

(15 * 9) / (1 * 5) = 135 / 5

Now we need to simplify the fraction 135/5. Both 135 and 5 are divisible by 5, so we can divide both the numerator and the denominator by 5:

135 / 5 = 27 5 / 5 = 1

So, 135/5 simplifies to 27/1, which is just 27. Therefore, our equation now reads:

x = 27

And there you have it! We've successfully solved for x. The value of x that makes the equation (2/3)x - (1/9)x + 5 = 20 true is 27.

This final step of solving for 'x' highlights the power of using inverse operations to unravel equations. It's like carefully dismantling a puzzle, piece by piece, until you reveal the hidden solution. Each inverse operation we perform brings us closer to isolating the variable and uncovering its value.

But solving for 'x' isn't just about finding a number that satisfies the equation. It's about understanding the underlying relationships between the different parts of the equation. It's about seeing how changing one part of the equation affects the others and how we can manipulate those relationships to our advantage. This understanding is what truly empowers you to solve more complex problems and tackle new challenges in mathematics and beyond.

Furthermore, the process we've followed – combining like terms, isolating the variable, and using inverse operations – is a general strategy that can be applied to a wide range of algebraic equations. It's a recipe for success that you can use again and again, with slight variations depending on the specific equation you're facing.

So, as you continue your mathematical journey, remember these steps and practice applying them to different types of equations. The more you practice, the more confident and skilled you'll become at solving for 'x' and other variables. You'll start to see patterns and develop an intuition for how to approach different problems. And who knows, maybe one day you'll even be teaching others how to solve equations just like this one!

Conclusion

Alright, guys, we did it! We successfully solved the equation (2/3)x - (1/9)x + 5 = 20 and found that x = 27. We broke it down into easy-to-follow steps: combining like terms, isolating the x term, and finally, solving for x. Remember these steps, and you'll be able to tackle similar equations with confidence. Keep practicing, and you'll become a true equation-solving master!

Solving equations is a fundamental skill in mathematics, and it's a skill that you'll use in many different contexts, both in and out of the classroom. It's not just about manipulating symbols on a page; it's about developing logical thinking, problem-solving abilities, and a deeper understanding of how mathematical relationships work.

Think about it – equations are essentially mathematical stories. They tell us how different quantities are related to each other. And when we solve an equation, we're essentially figuring out the missing piece of the story. We're using the information we have to uncover something that was previously unknown. This is a powerful skill, not just in mathematics, but in life in general.

Every time you encounter a problem, whether it's figuring out how much time you need to get somewhere, how much money you can spend, or how to optimize a process, you're essentially solving an equation. You're taking the information you have, identifying the unknowns, and using logical steps to find the solution.

So, by mastering the art of equation solving, you're not just learning a mathematical technique; you're developing a mindset. You're learning how to approach problems systematically, how to break them down into smaller parts, and how to use logic and reasoning to find the answers. These are skills that will serve you well in any field you choose to pursue.

And remember, the key to mastering any skill is practice. The more you solve equations, the more comfortable and confident you'll become. You'll start to see patterns and develop your own strategies for tackling different types of problems. You'll learn from your mistakes and grow stronger with each challenge you overcome.

So, don't be afraid to embrace the challenge of solving equations. See them as puzzles to be solved, stories to be told, and opportunities to learn and grow. With practice and perseverance, you'll become a true equation-solving expert!