Solving The Equation (-22+3x)/(3x+7)=2 A Step-by-Step Guide

Hey guys! Let's dive into solving equations, a fundamental skill in mathematics. Equations might seem intimidating at first, but breaking them down into manageable steps makes the process much clearer. In this article, we'll tackle the equation $\frac{-22+3x}{3x+7} = 2$, showing you how to isolate the variable x and find the solution. Think of it like solving a puzzle – each step gets you closer to the final answer!

Understanding the Equation

Before we jump into solving, let's take a moment to understand the equation we're working with: $\frac{-22+3x}{3x+7} = 2$. This equation involves a fraction where the numerator is $-22 + 3x$ and the denominator is $3x + 7$. Our goal is to get x by itself on one side of the equation. This involves undoing the operations that are currently being applied to x, working backward from the order of operations (PEMDAS/BODMAS). The key here is to maintain balance – whatever operation you perform on one side of the equation, you must perform on the other side to keep the equation true. It's like a seesaw; you need to keep both sides level!

Step 1 Clearing the Fraction – Multiplying Both Sides

The first hurdle in solving this equation is the fraction. Fractions can look scary, but we can easily get rid of them by multiplying both sides of the equation by the denominator. In our case, the denominator is $3x + 7$. So, we multiply both sides of the equation $\frac{-22+3x}{3x+7} = 2$ by $(3x + 7)$. This gives us: $(3x + 7) * \frac{-22+3x}{3x+7} = 2 * (3x + 7)$. On the left side, the $(3x + 7)$ in the numerator and denominator cancel each other out, leaving us with $-22 + 3x$. On the right side, we need to distribute the 2 across the terms inside the parentheses. This means multiplying 2 by both $3x$ and 7. So, the equation becomes: $-22 + 3x = 6x + 14$. See? No more fraction! We've made the equation much simpler to work with. This step is crucial because it transforms a complex equation into a more familiar linear equation. Remember, multiplying both sides by the denominator is a standard technique for clearing fractions in equations.

Step 2 Isolating the x Terms – Moving Variables to One Side

Now that we've cleared the fraction, the next step is to gather all the terms containing x on one side of the equation. We have x terms on both sides: $3x$ on the left and $6x$ on the right. To get all the x terms on one side, we need to eliminate the x term from one of the sides. A common strategy is to subtract the smaller x term from both sides. In this case, $3x$ is smaller than $6x$, so we'll subtract $3x$ from both sides of the equation $-22 + 3x = 6x + 14$. This gives us: $-22 + 3x - 3x = 6x + 14 - 3x$. Simplifying both sides, we get: $-22 = 3x + 14$. Notice that the $3x$ term on the left side has been eliminated, and we now have all the x terms on the right side. This step is all about rearranging the equation to make it easier to isolate x. By moving the x terms to one side, we're one step closer to solving for x.

Step 3 Isolating the Constant Terms – Moving Numbers to the Other Side

We're making great progress! Our equation is now $-22 = 3x + 14$. We have the x term on the right side, so now we need to isolate the x term by moving all the constant terms (the numbers without x) to the left side. Currently, we have $+14$ on the right side. To eliminate it, we need to perform the inverse operation, which is subtraction. We subtract 14 from both sides of the equation: $-22 - 14 = 3x + 14 - 14$. Simplifying both sides, we get: $-36 = 3x$. Now, we have all the constant terms on the left side and just the x term on the right side. This is exactly what we want! We're almost there. This step demonstrates the power of inverse operations in solving equations. By using subtraction to undo addition, we're systematically isolating the variable.

Step 4 Solving for x – Dividing to Isolate the Variable

We're in the home stretch! Our equation is now $-36 = 3x$. The only thing standing between us and the solution is the coefficient 3 that's multiplying x. To isolate x, we need to undo this multiplication. The inverse operation of multiplication is division, so we divide both sides of the equation by 3: $\frac{-36}{3} = \frac{3x}{3}$. Simplifying both sides, we get: $-12 = x$. And there you have it! We've solved for x. The solution to the equation is $x = -12$. This final step highlights the fundamental principle of solving equations: using inverse operations to isolate the variable. By dividing both sides by the coefficient, we've successfully found the value of x that makes the equation true.

Step 5 Verification – Plugging the Solution Back into the Original Equation

It's always a good idea to check your answer to make sure it's correct. This is especially important in equations that involve fractions or other complexities. To verify our solution, we substitute $x = -12$ back into the original equation, $\frac{-22+3x}{3x+7} = 2$, and see if both sides of the equation are equal. Substituting $x = -12$, we get: $\frac{-22 + 3(-12)}{3(-12) + 7} = 2$. Now, let's simplify: $\frac{-22 - 36}{-36 + 7} = 2$. This becomes: $\frac{-58}{-29} = 2$. And finally: $2 = 2$. The left side of the equation equals the right side, so our solution $x = -12$ is correct! Verification is a crucial step in the problem-solving process. It gives you confidence in your answer and helps you catch any potential errors. By plugging the solution back into the original equation, you can confirm that it satisfies the equation and that you haven't made any mistakes along the way.

Final Answer – The Solution and Summary

So, guys, we've successfully navigated the equation $\frac{-22+3x}{3x+7} = 2$! We started by clearing the fraction, then systematically isolated the x terms and the constant terms. Finally, we divided to solve for x and verified our solution. The solution to the equation is $x = -12$. Remember, solving equations is a step-by-step process. By breaking down the equation into smaller, manageable steps, you can tackle even the most complex problems. The key is to understand the operations being performed on the variable and to use inverse operations to undo them. And always remember to verify your solution to ensure accuracy. Keep practicing, and you'll become a pro at solving equations in no time!

Key Takeaways for Mastering Equation Solving

Solving equations is a cornerstone of mathematics, and mastering it opens doors to more advanced concepts. To solidify your understanding, let's recap the key takeaways from our journey with the equation $\frac{-22+3x}{3x+7} = 2$.

First and foremost, always aim to isolate the variable. This means getting the variable x (or whatever variable you're solving for) by itself on one side of the equation. This is the overarching goal that guides every step you take. Think of it as the destination on your mathematical map.

Next, master the art of inverse operations. Each mathematical operation has an inverse that undoes it: addition and subtraction are inverses, as are multiplication and division. When solving equations, you use inverse operations to peel away the layers surrounding the variable, like unwrapping a present. For instance, if a number is being added to the variable, you subtract that number from both sides.

Clearing fractions is a game-changer. Fractions can often make equations look intimidating, but they're easily handled by multiplying both sides of the equation by the denominator. This eliminates the fraction and transforms the equation into a more manageable form. It's like switching from a bumpy dirt road to a smooth highway.

Maintaining balance is paramount. Equations are like scales; they must remain balanced. Any operation you perform on one side of the equation must also be performed on the other side to maintain equality. This principle ensures that you're not changing the solution, just rearranging the equation to reveal it.

Verification is your safety net. Plugging your solution back into the original equation is the ultimate way to check your work. If both sides of the equation are equal, you've found the correct solution. If not, you know there's a mistake somewhere, and you can go back and review your steps. It's like having a spellchecker for your math.

Finally, practice makes perfect. Solving equations is a skill that improves with practice. The more equations you solve, the more comfortable you'll become with the process, and the quicker you'll be able to identify the best approach. Think of it like learning a musical instrument; the more you practice, the more fluent you become.

By keeping these key takeaways in mind, you'll be well-equipped to tackle a wide range of equations with confidence and precision. So, keep practicing, keep exploring, and enjoy the journey of mathematical discovery!

Common Pitfalls to Avoid When Solving Equations

Even with a clear understanding of the steps involved, it's easy to stumble when solving equations. Let's highlight some common pitfalls that students often encounter, so you can steer clear of them and boost your equation-solving prowess.

One frequent mistake is forgetting to distribute properly. When multiplying a number or variable by an expression in parentheses, you must distribute it to every term inside the parentheses. Failing to do so can lead to incorrect solutions. Imagine you're baking a cake, and you forget to add sugar to all the batter – the cake won't taste right! Similarly, missing a term during distribution throws off the entire equation.

Another common error is combining like terms incorrectly. Only terms with the same variable and exponent can be combined. For instance, $3x$ and $5x$ can be combined, but $3x$ and $5x^2$ cannot. Mixing unlike terms is like trying to add apples and oranges – they're different things!

Incorrectly applying inverse operations is another pitfall. Remember, the goal is to isolate the variable by undoing the operations that are being applied to it. Make sure you're using the correct inverse operation – subtraction to undo addition, division to undo multiplication, and so on. Using the wrong operation is like trying to unlock a door with the wrong key – it just won't work.

Forgetting to perform the same operation on both sides is a critical error. The principle of balance is fundamental to solving equations. Whatever you do to one side, you must do to the other. Failing to maintain balance is like tilting a seesaw too far to one side – it throws everything off.

Rushing through the steps is a temptation, especially when you feel confident. However, rushing can lead to careless errors. Take your time, write out each step clearly, and double-check your work. It's like proofreading an essay before submitting it – a little extra care can catch mistakes you might otherwise miss.

Finally, skipping the verification step is a risky move. As we discussed earlier, plugging your solution back into the original equation is the best way to confirm its accuracy. Skipping this step is like submitting a puzzle without checking if the pieces fit – you might be missing something important.

By being aware of these common pitfalls and taking steps to avoid them, you'll significantly improve your equation-solving skills and achieve greater accuracy. Remember, solving equations is a journey, and every mistake is a learning opportunity. So, embrace the challenges, learn from your errors, and keep striving for mastery!

Practice Problems to Sharpen Your Skills

Now that we've explored the ins and outs of solving equations, it's time to put your knowledge to the test! Practice is the key to solidifying your understanding and building confidence. Let's dive into some practice problems that will challenge you and help you hone your equation-solving skills. Remember, the goal is not just to find the answer, but also to understand the process.

Problem 1: Solve for y: $5y - 3 = 12$. This is a classic two-step equation that requires you to isolate y by using inverse operations. Think about what operations are being applied to y and what inverse operations you need to perform to undo them. Don't forget to maintain balance by performing the same operation on both sides of the equation.

Problem 2: Solve for a: $2(a + 4) = 18$. This equation involves distribution, so make sure you multiply the 2 by both terms inside the parentheses. After distributing, you'll have a two-step equation that you can solve using inverse operations.

Problem 3: Solve for m: $\frac{m}{3} + 2 = 7$. This equation involves a fraction, so the first step is to clear the fraction by multiplying both sides of the equation by the denominator. Once you've cleared the fraction, you can solve the equation using inverse operations.

Problem 4: Solve for x: $4x + 5 = 2x - 1$. This equation has x terms on both sides, so you'll need to gather the x terms on one side of the equation before you can isolate x. Remember to subtract the smaller x term from both sides to avoid dealing with negative coefficients.

Problem 5: Solve for z: $\frac{-15 + 2z}{5z + 10} = 3$. This equation combines fractions and variables in both the numerator and denominator. The first step is to clear the fraction by multiplying both sides of the equation by the denominator. After that, you'll need to distribute, gather like terms, and isolate z. This problem is a great way to test your comprehensive understanding of equation-solving techniques.

As you work through these problems, remember to show your work clearly and systematically. Writing out each step will help you track your progress, identify any errors, and develop a logical approach to solving equations. And don't forget to verify your solutions by plugging them back into the original equations!

If you encounter any difficulties, don't get discouraged. Review the key concepts and techniques we've discussed in this article, and try breaking the problem down into smaller, more manageable steps. With practice and persistence, you'll master the art of solving equations and unlock a whole new world of mathematical possibilities. So, grab your pencil, fire up your brain, and let's get solving!

Conclusion Unleashing Your Equation-Solving Potential

Congratulations on making it to the end of this comprehensive guide on solving equations! We've covered a lot of ground, from the fundamental principles of isolating variables to common pitfalls to avoid and practice problems to sharpen your skills. By now, you should have a solid understanding of the equation-solving process and the confidence to tackle a wide range of problems.

Solving equations is not just a mathematical skill; it's a powerful tool that can be applied in many areas of life. From balancing your budget to planning a trip, the ability to solve equations can help you make informed decisions and achieve your goals. It's like having a superpower that allows you to unravel complex situations and find the solutions you need.

But the journey doesn't end here. The world of mathematics is vast and fascinating, and there's always more to learn. So, continue to explore, continue to practice, and continue to challenge yourself. The more you engage with mathematics, the more you'll discover its beauty and power.

Remember, the key to success in mathematics is not just memorizing formulas and procedures, but also developing a deep understanding of the underlying concepts. Focus on the "why" behind the "what," and you'll find that mathematics becomes not just a subject to study, but a way of thinking and a way of seeing the world.

So, go forth and unleash your equation-solving potential! Embrace the challenges, celebrate your successes, and never stop learning. The world needs problem-solvers, and you have the potential to be one of them. Keep practicing, keep exploring, and keep shining your mathematical light!