Solve 6z+1/2+4x=3x/5+9-7x/3: Step-by-Step Guide

Hey guys! Today, we're diving deep into the exciting world of equation solving! We'll be tackling a specific problem: 6z + 1/2 + 4x = 3x/5 + 9 - 7x/3. Don't worry if it looks intimidating at first glance; we'll break it down step by step, making it super easy to understand. Solving equations is a fundamental skill in mathematics, crucial for various fields, from engineering to economics. Mastering it opens doors to more advanced concepts and real-world problem-solving. So, grab your pencils, and let's get started on this mathematical adventure!

Understanding the Basics of Equation Solving

Before we jump into the main equation, let's quickly recap the basic principles of equation solving. An equation is essentially a mathematical statement asserting the equality of two expressions. Our goal is to find the value(s) of the variable(s) that make this statement true. We achieve this by manipulating the equation while maintaining the equality. The key principle here is: whatever operation you perform on one side of the equation, you must perform the same operation on the other side. Think of it like a balance scale – to keep it balanced, you need to add or subtract the same weight from both sides.

Common operations include addition, subtraction, multiplication, and division. We can also simplify expressions by combining like terms, distributing constants, and using the order of operations (PEMDAS/BODMAS). Remember, the ultimate aim is to isolate the variable we're solving for on one side of the equation. This process often involves multiple steps, each carefully designed to bring us closer to the solution. Keep in mind that some equations might have one solution, no solutions, or infinitely many solutions. The nature of the equation dictates the type and number of solutions we can expect. So, with these foundational concepts in mind, let's move on to our specific equation and see how these principles apply in practice. We'll be using these techniques extensively as we work through our example, so it's vital to have a solid grasp of them. Remember, practice makes perfect, so don't hesitate to work through similar problems to reinforce your understanding!

Step-by-Step Solution of 6z + 1/2 + 4x = 3x/5 + 9 - 7x/3

Okay, let's tackle our equation: 6z + 1/2 + 4x = 3x/5 + 9 - 7x/3. The first thing we notice is the presence of fractions. Fractions can sometimes make equations look more complicated than they are, so our initial strategy will be to eliminate them. To do this, we need to find the least common multiple (LCM) of the denominators. In this case, we have denominators 2, 5, and 3. The LCM of 2, 5, and 3 is 30. So, we'll multiply both sides of the equation by 30. This crucial step will clear the fractions and give us a more manageable equation to work with. Remember, multiplying both sides by the same number maintains the equality, a fundamental principle in equation solving.

Multiplying both sides by 30, we get:

30 * (6z + 1/2 + 4x) = 30 * (3x/5 + 9 - 7x/3)

Now, we distribute the 30 on both sides:

(30 * 6z) + (30 * 1/2) + (30 * 4x) = (30 * 3x/5) + (30 * 9) - (30 * 7x/3)

This simplifies to:

180z + 15 + 120x = 18x + 270 - 70x

Great! We've successfully eliminated the fractions. Now, let's focus on combining like terms. On the left side, we have 180z, 15, and 120x. On the right side, we have 18x, 270, and -70x. We can combine the x terms on the right side: 18x - 70x = -52x. So, the equation now looks like this:

180z + 15 + 120x = -52x + 270

Next, we want to group the x terms together. Let's add 52x to both sides of the equation:

180z + 15 + 120x + 52x = -52x + 270 + 52x

This simplifies to:

180z + 15 + 172x = 270

Now, let's isolate the terms with variables on one side and the constants on the other. We'll subtract 15 from both sides:

180z + 15 + 172x - 15 = 270 - 15

This gives us:

180z + 172x = 255

At this point, we have an equation with two variables, z and x. To find specific solutions for z and x, we would need another equation relating these variables. However, without additional information, we can express one variable in terms of the other. For example, we can solve for z in terms of x, or vice versa. Let's solve for z in terms of x. To do this, we'll subtract 172x from both sides:

180z + 172x - 172x = 255 - 172x

This simplifies to:

180z = 255 - 172x

Finally, we'll divide both sides by 180 to isolate z:

z = (255 - 172x) / 180

This is our solution for z in terms of x. We can simplify this fraction further by finding the greatest common divisor (GCD) of 255, 172, and 180. The GCD is 1. So, we can't simplify the fraction further in terms of reducing the coefficients. However, we can leave the solution in this form: z = (255 - 172x) / 180. This means that for any value we choose for x, we can plug it into this equation and find the corresponding value of z. This equation represents a family of solutions, rather than a single unique solution, because we have two variables and only one equation. If we were given another equation relating z and x, we could then solve for specific values of z and x that satisfy both equations simultaneously. So, we've successfully solved for z in terms of x. Remember, the key steps were eliminating fractions, combining like terms, and isolating the variable we wanted to solve for. These techniques are fundamental to solving various types of equations, so mastering them is essential for your mathematical journey!

Dealing with Two Variables: Expressing One in Terms of the Other

As we saw in the previous section, our equation 180z + 172x = 255 has two variables, z and x. This means we can't find unique numerical values for both variables with just this one equation. Instead, we express one variable in terms of the other, which gives us a relationship between them. This relationship represents an infinite number of solutions, each satisfying the original equation. Think of it like a line on a graph – each point on the line represents a pair of (x, z) values that makes the equation true. To find specific values for x and z, we would need another independent equation involving these same variables. This is the basis of solving systems of equations, a topic we might explore in the future.

For now, let's reiterate the importance of expressing one variable in terms of the other. It's a powerful technique that allows us to understand the relationship between variables even when we can't pinpoint exact numerical solutions. This is particularly useful in real-world applications where relationships between quantities are more important than specific values. For example, in economics, we might want to understand how the price of a product affects its demand, even if we don't know the exact price or demand at a given time. Expressing demand in terms of price gives us valuable insights into this relationship. Similarly, in physics, we might want to understand how the position of an object changes with time, even if we don't know its exact position at a particular moment. Expressing position as a function of time allows us to analyze the object's motion. So, mastering the technique of expressing one variable in terms of the other is a valuable skill that extends far beyond the realm of pure mathematics. It's a tool that can help us understand and analyze relationships in various fields of study.

Common Mistakes and How to Avoid Them

Alright, let's talk about some common pitfalls that people often encounter when solving equations, especially ones like 6z + 1/2 + 4x = 3x/5 + 9 - 7x/3. Knowing these mistakes can help you steer clear of them and boost your accuracy. One frequent error is incorrect distribution. Remember, when multiplying a number by an expression in parentheses, you must multiply it by each term inside the parentheses. For example, when we multiplied both sides of the equation by 30, we had to multiply 30 by 6z, 1/2, and 4x on the left side, and by 3x/5, 9, and -7x/3 on the right side. Forgetting to multiply by even one term can throw off the entire solution.

Another common mistake is mishandling signs, especially when dealing with negative numbers. Pay close attention to the signs when combining like terms and moving terms from one side of the equation to the other. A simple sign error can lead to a completely wrong answer. Double-checking your work and paying extra attention to negative signs can significantly reduce this type of error. Failing to find the correct least common multiple (LCM) when clearing fractions is another potential pitfall. If you use the wrong LCM, you'll still eliminate the fractions, but you'll end up with larger coefficients, making the equation more difficult to work with. Make sure you understand how to find the LCM correctly before tackling equations with fractions.

Also, remember the golden rule of equation solving: whatever you do to one side, you must do to the other. This principle ensures that the equality is maintained throughout the solution process. Neglecting this rule will inevitably lead to incorrect results. Finally, don't forget to simplify your answer as much as possible. This might involve combining like terms, reducing fractions, or expressing the solution in a more concise form. A simplified answer is not only easier to understand but also reduces the chance of making errors in subsequent calculations. By being aware of these common mistakes and taking steps to avoid them, you can become a more confident and accurate equation solver. Remember, practice makes perfect, so keep working through problems and refining your skills!

Tips and Tricks for Mastering Equation Solving

Now that we've gone through the solution and discussed common mistakes, let's delve into some tips and tricks that can help you master equation solving. These strategies will not only improve your accuracy but also make the process more efficient and enjoyable. First and foremost, always simplify each side of the equation as much as possible before you start moving terms around. This means combining like terms, distributing constants, and simplifying fractions. By simplifying early on, you'll reduce the complexity of the equation and make subsequent steps easier to manage.

Next, develop a systematic approach to solving equations. This might involve following a specific set of steps, such as clearing fractions, combining like terms, isolating the variable, and simplifying the result. Having a consistent approach will help you stay organized and avoid making careless errors. Another valuable tip is to check your solution by plugging it back into the original equation. If the solution is correct, it should make the equation true. This is a great way to verify your work and catch any mistakes you might have made. If the solution doesn't check out, go back and review your steps to identify the error.

When dealing with equations involving fractions, remember the technique of multiplying both sides by the least common multiple (LCM) of the denominators. This will eliminate the fractions and give you a simpler equation to work with. If you're unsure how to find the LCM, review this concept – it's a crucial skill for equation solving. Don't be afraid to use scratch paper to work through intermediate steps. Trying to do everything in your head can lead to errors, especially when dealing with complex equations. Writing down each step clearly will help you stay organized and track your progress.

Finally, remember that practice is key to mastering equation solving. The more problems you solve, the more comfortable and confident you'll become. Start with simpler equations and gradually work your way up to more challenging ones. Seek out additional resources, such as textbooks, online tutorials, and practice problems, to further enhance your skills. With consistent effort and the right strategies, you can conquer any equation that comes your way! So, keep practicing, keep learning, and most importantly, keep enjoying the process of mathematical discovery!

Conclusion: The Power of Equation Solving

Wow, we've covered a lot in this article! We took on the equation 6z + 1/2 + 4x = 3x/5 + 9 - 7x/3, broke it down step by step, and explored the key concepts and techniques involved in solving it. We discussed the importance of eliminating fractions, combining like terms, and isolating variables. We also addressed common mistakes and provided tips and tricks for mastering equation solving. But more than just solving this specific equation, we've gained a deeper understanding of the power and versatility of equation solving as a fundamental mathematical skill.

Equation solving is not just about manipulating symbols and numbers; it's about developing logical thinking, problem-solving skills, and a systematic approach to challenges. These skills are valuable not only in mathematics but also in various aspects of life, from everyday decision-making to complex scientific endeavors. The ability to analyze a problem, identify the relevant information, and apply appropriate strategies is a hallmark of critical thinking, and equation solving is an excellent way to hone this ability.

Furthermore, equation solving forms the foundation for many advanced mathematical concepts, such as algebra, calculus, and differential equations. Mastering these concepts opens doors to a wide range of fields, including engineering, physics, computer science, economics, and finance. So, by investing time and effort in mastering equation solving, you're not just learning a mathematical skill; you're building a foundation for future success in various academic and professional pursuits. Remember, the journey of learning mathematics is a continuous process. There will be challenges along the way, but with persistence, practice, and a positive attitude, you can overcome any obstacle. So, keep exploring, keep questioning, and keep solving! The world of mathematics is vast and fascinating, and equation solving is just one of the many exciting adventures that await you.