Solving Equations A Step By Step Guide

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Hey guys! Today, we're diving deep into solving equations, specifically the one you've got there: 23−4x+72=−9x+56\frac{2}{3}-4x+\frac{7}{2}=-9x+\frac{5}{6}. Don't worry if it looks a bit intimidating with all those fractions – we're going to break it down step by step, so it's super easy to follow. Solving equations is a fundamental skill in mathematics, and mastering it will open doors to more advanced concepts. This guide aims to provide a clear, conversational approach to understanding and solving this type of equation. We'll cover everything from the basic principles to the specific steps needed to find the value of 'x'. So, grab your thinking caps, and let's get started!

1. Understanding the Basics of Equations

Before we jump into solving the equation, let's make sure we're all on the same page with the basics. An equation is a mathematical statement that shows two expressions are equal. Think of it like a balanced scale – whatever you do to one side, you must do to the other to keep it balanced. Our goal in solving an equation is to isolate the variable (in this case, 'x') on one side of the equation. This means we want to get 'x' all by itself, so we know its value. To achieve this, we use inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. By applying these operations strategically, we can simplify the equation and eventually find the solution. Remember, the key is to keep the equation balanced throughout the process. This means performing the same operation on both sides of the equals sign. Whether you are dealing with simple linear equations or more complex algebraic expressions, the fundamental principle remains the same: isolate the variable by using inverse operations. Understanding this basic principle will make solving any equation much more manageable.

2. Clearing the Fractions: A Smart First Step

Fractions can sometimes make equations look messier than they actually are. A neat trick to simplify things is to clear the fractions right at the beginning. We do this by finding the least common multiple (LCM) of the denominators. In our equation, 23−4x+72=−9x+56\frac{2}{3}-4x+\frac{7}{2}=-9x+\frac{5}{6}, the denominators are 3, 2, and 6. The LCM of these numbers is 6. Why 6? Because 6 is the smallest number that all three denominators (3, 2, and 6) divide into evenly. Now, we're going to multiply every single term in the equation by 6. This is crucial – you need to multiply every term, both on the left-hand side and the right-hand side, to keep the equation balanced. When we multiply each term by 6, the fractions will magically disappear! This is because the denominators will cancel out with the 6. For example, when we multiply 23\frac{2}{3} by 6, we get 123\frac{12}{3}, which simplifies to 4. Similarly, when we multiply 72\frac{7}{2} by 6, we get 422\frac{42}{2}, which simplifies to 21. And when we multiply 56\frac{5}{6} by 6, we get 306\frac{30}{6}, which simplifies to 5. This step is super helpful because it transforms the equation into one with whole numbers, which are much easier to work with. By clearing the fractions, we eliminate a potential source of error and make the equation more manageable, setting us up for a smoother solving process. After this step, the equation will be in a much friendlier form, ready for the next stages of simplification.

3. Applying the Multiplication and Simplifying

Okay, let's put that LCM trick into action! We're going to multiply both sides of the equation 23−4x+72=−9x+56\frac{2}{3}-4x+\frac{7}{2}=-9x+\frac{5}{6} by 6. Remember, we need to multiply each term by 6. So, we have:

6 * (23\frac{2}{3}) - 6 * (4x) + 6 * (72\frac{7}{2}) = 6 * (-9x) + 6 * (56\frac{5}{6})

Now, let's simplify each term. 6 multiplied by 23\frac{2}{3} is 4 (because 6/3 is 2, and 22 is 4). 6 multiplied by -4x is -24x. 6 multiplied by 72\frac{7}{2} is 21 (because 6/2 is 3, and 37 is 21). On the right side, 6 multiplied by -9x is -54x, and 6 multiplied by 56\frac{5}{6} is 5 (because the 6s cancel out). So, after multiplying and simplifying, our equation now looks like this:

4 - 24x + 21 = -54x + 5

See how much cleaner it looks without those fractions? Now, we can combine like terms on the left side. We have 4 and 21, which add up to 25. So, our equation simplifies further to:

25 - 24x = -54x + 5

This simplified equation is much easier to handle. We've successfully eliminated the fractions and combined the constants on one side. The next step involves getting all the 'x' terms on one side and the constants on the other. This process of applying the multiplication and simplifying is crucial in making the equation more manageable and setting us up for the next steps in solving for 'x'. This methodical approach ensures we're moving closer to the solution with each step.

4. Isolating the Variable: Bringing 'x' to One Side

Now that we have a simplified equation, 25 - 24x = -54x + 5, it's time to isolate the variable 'x'. This means we want to get all the terms with 'x' on one side of the equation and all the constant terms (the numbers without 'x') on the other side. A common strategy is to move the 'x' terms to the side where the coefficient of 'x' is larger. In our case, -24x is greater than -54x (remember, with negative numbers, the one closer to zero is larger). So, let's move the -54x term to the left side of the equation. To do this, we add 54x to both sides (remember, whatever we do to one side, we must do to the other to keep the equation balanced). Adding 54x to both sides gives us:

25 - 24x + 54x = -54x + 54x + 5

Simplifying this, -24x + 54x becomes 30x, and -54x + 54x cancels out to zero. So, our equation now looks like this:

25 + 30x = 5

Great! We've successfully moved the 'x' terms to the left side. Now, let's move the constant term, 25, to the right side. To do this, we subtract 25 from both sides:

25 - 25 + 30x = 5 - 25

Simplifying, 25 - 25 is zero, and 5 - 25 is -20. So, our equation now looks like this:

30x = -20

We're getting closer! We've isolated the 'x' term on one side of the equation. The next and final step is to solve for 'x' by dividing both sides by the coefficient of 'x'.

5. Solving for 'x': The Final Step

We've reached the final stage of solving our equation! We've simplified it down to 30x = -20. To find the value of 'x', we need to isolate it completely. Since 'x' is being multiplied by 30, we need to perform the inverse operation, which is division. So, we'll divide both sides of the equation by 30. This is another crucial step where we maintain the balance of the equation by doing the same thing to both sides.

Dividing both sides by 30, we get:

30x / 30 = -20 / 30

On the left side, 30x divided by 30 simplifies to just 'x'. On the right side, -20 divided by 30 can be simplified. Both -20 and 30 are divisible by 10, so we can reduce the fraction. -20 divided by 10 is -2, and 30 divided by 10 is 3. Therefore, -20/30 simplifies to -2/3.

So, our solution is:

x = -2/3

We've done it! We've successfully solved the equation 23−4x+72=−9x+56\frac{2}{3}-4x+\frac{7}{2}=-9x+\frac{5}{6}. The value of 'x' that makes this equation true is -2/3. To ensure our solution is correct, we can substitute this value back into the original equation and check if both sides are equal. This process, known as verifying the solution, adds a layer of confidence to our answer. Solving for 'x' involves a systematic approach, and each step, from clearing fractions to isolating the variable, plays a critical role in reaching the correct answer.

6. Verifying the Solution: Ensuring Accuracy

It's always a good idea to double-check our work, especially in math! We can verify our solution by substituting the value we found for 'x' back into the original equation. In our case, we found that x = -2/3. So, let's plug this value into the equation 23−4x+72=−9x+56\frac{2}{3}-4x+\frac{7}{2}=-9x+\frac{5}{6} and see if both sides come out to be equal.

Substituting x = -2/3 into the equation, we get:

23−4(−23)+72=−9(−23)+56\frac{2}{3} - 4(-\frac{2}{3}) + \frac{7}{2} = -9(-\frac{2}{3}) + \frac{5}{6}

Now, let's simplify each side. On the left side, -4 multiplied by -2/3 is 8/3. So, the left side becomes:

23+83+72\frac{2}{3} + \frac{8}{3} + \frac{7}{2}

Combining the fractions with the same denominator, 23+83\frac{2}{3} + \frac{8}{3} is 103\frac{10}{3}. Now, we have:

103+72\frac{10}{3} + \frac{7}{2}

To add these fractions, we need a common denominator, which is 6. Converting the fractions, we get 206+216\frac{20}{6} + \frac{21}{6}, which equals 416\frac{41}{6}. So, the left side of the equation simplifies to 41/6.

Now, let's simplify the right side. -9 multiplied by -2/3 is 6. So, the right side becomes:

6 + 56\frac{5}{6}

Converting 6 to a fraction with a denominator of 6, we get 366\frac{36}{6}. So, we have:

366+56\frac{36}{6} + \frac{5}{6}

Adding these fractions, we get 416\frac{41}{6}. So, the right side of the equation also simplifies to 41/6.

Since both the left side and the right side of the equation equal 41/6 when we substitute x = -2/3, our solution is correct! Verifying the solution is an important step in problem-solving, as it ensures accuracy and helps in building confidence in your mathematical skills. This process not only confirms the answer but also deepens the understanding of the equation and its solution.

7. Key Takeaways and Tips for Solving Equations

Alright, guys, we've walked through solving the equation 23−4x+72=−9x+56\frac{2}{3}-4x+\frac{7}{2}=-9x+\frac{5}{6} step by step. Before we wrap up, let's highlight some key takeaways and tips that will help you tackle similar problems in the future:

  • Clear Fractions First: As we saw, getting rid of fractions early on can make the equation much easier to work with. Find the least common multiple (LCM) of the denominators and multiply every term in the equation by it.
  • Simplify Both Sides: Before you start moving terms around, simplify each side of the equation by combining like terms. This can prevent errors and make the equation more manageable.
  • Isolate the Variable: The goal is to get the variable (in this case, 'x') alone on one side of the equation. Use inverse operations (addition/subtraction, multiplication/division) to move terms to the correct side.
  • Keep the Equation Balanced: Remember the golden rule – whatever you do to one side of the equation, you must do to the other. This ensures that the equation remains balanced and the solution remains accurate.
  • Verify Your Solution: Always take the time to plug your solution back into the original equation to check if it's correct. This can catch any mistakes and give you confidence in your answer.

Solving equations is like learning a new language – it takes practice, but once you get the hang of it, it becomes much easier. Don't be afraid to make mistakes; they're part of the learning process. The more you practice, the more confident and proficient you'll become at solving equations. So, keep practicing, stay patient, and remember these tips, and you'll be solving equations like a pro in no time!

8. Practice Problems: Sharpen Your Skills

Now that we've gone through the solution and key strategies, the best way to solidify your understanding is through practice. Here are a few similar equations for you to try solving on your own. Remember to follow the steps we discussed: clear fractions, simplify both sides, isolate the variable, and verify your solution. Practice makes perfect, and working through these problems will help you build confidence and skill in solving equations.

  1. 12+3x−54=2x+18\frac{1}{2} + 3x - \frac{5}{4} = 2x + \frac{1}{8}
  2. 35−2x+12=−5x+710\frac{3}{5} - 2x + \frac{1}{2} = -5x + \frac{7}{10}
  3. 43+6x−29=4x−13\frac{4}{3} + 6x - \frac{2}{9} = 4x - \frac{1}{3}

Work through these equations step by step, and if you encounter any difficulties, revisit the methods and strategies we covered earlier. Solving these practice problems will not only reinforce your understanding but also help you identify any areas where you might need further clarification. Remember, the key to mastering equation-solving is consistent practice and a methodical approach. So, grab a pencil, work through these problems, and watch your skills improve! With each equation you solve, you'll become more comfortable and confident in your ability to tackle even the most challenging mathematical problems.

9. Conclusion: Mastering Equations for Mathematical Success

Congratulations, guys! You've made it through this comprehensive guide on solving the equation 23−4x+72=−9x+56\frac{2}{3}-4x+\frac{7}{2}=-9x+\frac{5}{6}. You've learned the importance of clearing fractions, simplifying equations, isolating the variable, and verifying your solution. More importantly, you've gained valuable skills and strategies that you can apply to solve a wide range of equations. Mastering equations is a fundamental step in your mathematical journey. It's a skill that underpins many other mathematical concepts, from algebra to calculus and beyond. The ability to confidently solve equations opens doors to more advanced topics and problem-solving scenarios.

Remember, the key to success in mathematics is consistent practice and a clear understanding of the fundamental principles. Don't be discouraged by challenges; instead, view them as opportunities to learn and grow. The more you practice, the more intuitive equation-solving will become. By applying the steps and tips we've discussed, you'll be well-equipped to tackle any equation that comes your way.

So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. With dedication and the right approach, you can achieve anything you set your mind to. Happy solving!