Solving $6(x-6)-4=32x-118$ A Step-by-Step Guide
Breaking Down the Equation: A Step-by-Step Guide
Alright, guys, let's dive into this equation: . At first glance, it might seem a bit intimidating, but don't worry! We're going to break it down step by step, making it super easy to understand and solve. Our main goal here is to isolate the variable x on one side of the equation so we can figure out its value. To do this, we'll be using some basic algebraic principles like the distributive property, combining like terms, and inverse operations. So, grab your pencils, and let's get started!
Step 1: Distribute the 6
The first thing we need to tackle is the term . This means we need to distribute the 6 to both terms inside the parentheses. Remember the distributive property? It's where you multiply the term outside the parentheses by each term inside. So, we have:
So, becomes . Now, let's rewrite our equation with this simplification:
See? We're already making progress! The equation looks a little cleaner now.
Step 2: Combine Like Terms
Next up, let's simplify the left side of the equation further by combining like terms. We have two constant terms: -36 and -4. We can simply add these together:
So, our equation now looks like this:
Much better! We've reduced the number of terms on the left side, making it easier to work with.
Step 3: Move the x Terms to One Side
Now, we need to get all the x terms on one side of the equation. It doesn't matter which side we choose, but let's aim to keep the coefficient of x positive if we can, just to make things a little simpler down the line. In this case, we have on the left and on the right. To move the to the right side, we'll subtract from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced!
This simplifies to:
Great! Now all our x terms are on the right side.
Step 4: Move the Constants to the Other Side
We're getting closer! Now we need to isolate the x term by moving all the constant terms to the left side. We have -118 on the right side, so to move it to the left, we'll add 118 to both sides:
This simplifies to:
Looking good! We've managed to separate the x term from the constants.
Step 5: Isolate x
Finally, the last step! To isolate x, we need to get rid of the coefficient 26 that's multiplying it. To do this, we'll divide both sides of the equation by 26:
This gives us:
Or, written the other way around:
And there we have it! We've successfully solved the equation. The value of x that makes the equation true is 3.
Step 6: Verification (Always a Good Idea!)
To be absolutely sure we've got the right answer, it's always a good idea to plug our solution back into the original equation and see if it holds true. Let's substitute into our original equation:
Let's simplify both sides:
Yay! The equation holds true. This confirms that our solution, , is indeed correct. Always double-check, guys; it's a lifesaver!
Common Mistakes to Avoid When Solving Equations
Solving equations can sometimes feel like navigating a maze, and it's easy to stumble if you're not careful. To help you avoid some common pitfalls, let's go over a few mistakes that students often make. Being aware of these mistakes can help you approach equations with more confidence and accuracy.
Mistake 1: Forgetting to Distribute Properly
Remember the distributive property? It's a key part of solving many equations, especially those with parentheses. The mistake here is not multiplying the term outside the parentheses by every term inside. For instance, in our equation , we had to distribute the 6 to both x and -6. Forgetting to multiply the 6 by -6 would lead to an incorrect equation and, ultimately, the wrong solution. Always double-check that you've distributed correctly!
Mistake 2: Incorrectly Combining Like Terms
Combining like terms is another area where errors can easily creep in. The trick is to make sure you're only combining terms that are actually alike. For example, you can combine constants with constants and x terms with x terms, but you can't combine a constant with an x term. In our equation, we combined -36 and -4, which are both constants. Confusing these terms or making simple arithmetic errors during the addition or subtraction process can throw off your solution. Pay close attention to signs (positive and negative) as well!
Mistake 3: Not Performing Operations on Both Sides
One of the fundamental rules of equation-solving is that whatever you do to one side, you must do to the other. This keeps the equation balanced. A common mistake is adding, subtracting, multiplying, or dividing only on one side. For instance, when we subtracted from both sides of , we had to make sure to do it on both sides to maintain equality. Forgetting this rule is a surefire way to get the wrong answer.
Mistake 4: Sign Errors
Ah, sign errors โ the bane of many students' mathematical existence! These little buggers can sneak in when you're least expecting them, especially when dealing with negative numbers. For example, when moving terms across the equals sign, you need to change their signs. If you forget to do this, or if you make a mistake in adding or subtracting negative numbers, your solution will be off. It's a good idea to take extra care when handling negative numbers and to double-check your work.
Mistake 5: Skipping Steps
Sometimes, in an effort to save time, students try to skip steps when solving equations. While it's great to become efficient, skipping steps too early can lead to mistakes. Writing out each step, especially when you're first learning, helps you keep track of what you're doing and reduces the chance of errors. It also makes it easier to go back and check your work if something goes wrong. Remember, accuracy is more important than speed!
Mistake 6: Not Checking Your Solution
Last but definitely not least, one of the biggest mistakes is not checking your solution. It's like running a race and not looking back to see if you actually crossed the finish line! Plugging your solution back into the original equation is a simple yet powerful way to catch errors. If your solution doesn't make the equation true, you know you've made a mistake somewhere along the way, and you can go back and find it. It's a fantastic habit to get into.
By being mindful of these common mistakes, you can greatly improve your accuracy and confidence in solving equations. Keep practicing, and remember, even mathematicians make mistakes sometimes โ the key is to learn from them!
Real-World Applications of Solving Equations
Okay, so we've conquered the equation , and we've talked about how to avoid common mistakes. But you might be thinking, "Why does this even matter? When am I ever going to use this in real life?" Well, guys, solving equations is actually a super useful skill that pops up in all sorts of everyday situations. Let's take a look at some real-world applications to show you just how relevant this math stuff can be!
1. Budgeting and Finance
One of the most practical applications of solving equations is in budgeting and personal finance. Imagine you're trying to save up for a new gadget, a vacation, or maybe even a car. You need to figure out how much money you can save each month to reach your goal within a certain timeframe. This is where equations come in handy.
For example, let's say you want to save $1,200 in 6 months. You already have $200 saved up. You can set up an equation to find out how much more you need to save each month. If x represents the amount you need to save each month, the equation might look something like this:
Solving this equation will tell you exactly how much money you need to set aside each month to achieve your savings goal. Pretty cool, right? Whether it's planning your monthly expenses, calculating loan payments, or figuring out investment returns, solving equations is a crucial tool for managing your money wisely.
2. Cooking and Baking
Believe it or not, solving equations is even useful in the kitchen! Cooking and baking often involve adjusting recipes to make larger or smaller quantities. This means you need to scale the ingredients proportionally, which requires some basic algebra.
Let's say a recipe calls for 2 cups of flour to make 12 cookies, but you want to make 30 cookies. You can set up a proportion, which is essentially an equation, to figure out how much flour you need. If x represents the amount of flour needed for 30 cookies, the proportion might look like this:
Solving this equation will tell you exactly how many cups of flour you need. From doubling recipes for family gatherings to halving them for a smaller meal, understanding how to solve equations makes you a more versatile and confident cook.
3. Home Improvement Projects
Planning a home improvement project? Equations can help with that too! Whether you're building a fence, painting a room, or laying down new flooring, you'll need to calculate measurements, costs, and quantities. These calculations often involve setting up and solving equations.
For instance, let's say you want to build a rectangular garden with a specific perimeter. You know the length of one side, but you need to figure out the width. If w represents the width of the garden, and you know the length and the desired perimeter, you can create an equation to solve for w. Or, if you're trying to figure out how many gallons of paint you need to cover a wall, you'll need to calculate the area of the wall and then use an equation to determine the amount of paint required.
4. Travel and Navigation
Equations also play a role in travel and navigation. When you're planning a road trip, for example, you might want to estimate how long it will take to reach your destination. This involves using the formula distance = rate ร time, which is essentially an equation.
If you know the distance you need to travel and your average speed, you can solve for the time it will take. Or, if you have a specific amount of time to travel and you know the distance, you can solve for the speed you need to maintain. Similarly, equations can help you calculate fuel consumption, plan routes, and even convert between different units of measurement, like miles and kilometers.
5. Discounts and Sales
Who doesn't love a good deal? Solving equations is super handy when you're trying to figure out the sale price of an item or calculate discounts. Let's say a store is offering a 25% discount on a shirt that originally costs $40. You can use an equation to find out the sale price.
If x represents the sale price, you can set up the equation:
Solving this equation will tell you the discounted price of the shirt. Whether it's figuring out the final cost after a discount, calculating sales tax, or comparing prices, solving equations can help you make smart purchasing decisions and save money.
6. Time Management
Even managing your time effectively can involve solving equations. Let's say you have a busy day with lots of tasks to complete, and you want to allocate your time wisely. You can use equations to figure out how much time to spend on each task.
For example, if you have 3 hours (180 minutes) to study for two subjects, and you want to spend twice as much time on one subject as the other, you can set up an equation to determine how many minutes to allocate to each subject. If x represents the time spent on the first subject, the equation might look like this:
Solving this equation will tell you how much time to spend on each subject, helping you stay organized and productive.
As you can see, solving equations is a skill that's applicable in a wide range of real-life situations. From managing your finances to planning a trip, cooking a meal, or even just taking advantage of a sale, understanding how to set up and solve equations empowers you to make informed decisions and tackle practical problems with confidence. So, keep practicing, guys โ this is one math skill that will definitely come in handy!
Conclusion: Mastering Equations for a Brighter Future
Alright, guys, we've journeyed through the world of solving equations, tackled the problem , explored common mistakes to avoid, and discovered the countless real-world applications of this essential skill. By now, you should have a solid understanding of how to approach equations with confidence and accuracy. But before we wrap up, let's recap why mastering equations is so crucial and how it sets you up for success in various aspects of life.
Equations: The Foundation of Problem-Solving
At its core, solving equations is about problem-solving. It's about taking a complex situation, breaking it down into smaller, manageable parts, and using logical steps to find a solution. This problem-solving mindset extends far beyond the realm of mathematics. It's a skill that's invaluable in virtually every field and profession.
Whether you're a scientist conducting research, an engineer designing a bridge, a doctor diagnosing an illness, or a businessperson making strategic decisions, the ability to analyze a problem, identify key variables, and develop a systematic approach to finding a solution is paramount. And that's precisely what solving equations teaches you.
Equations: A Gateway to Higher Mathematics
Furthermore, mastering equations is a fundamental stepping stone to more advanced mathematical concepts. Algebra, calculus, statistics โ all these fields build upon the foundation of equation-solving skills. If you have a strong grasp of how to solve equations, you'll find it much easier to tackle more complex mathematical challenges down the road.
So, if you're aiming for a career in a STEM field (science, technology, engineering, and mathematics), a solid understanding of equations is non-negotiable. But even if you're not pursuing a STEM career, the analytical and logical thinking skills you develop by solving equations will serve you well in any profession.
Equations: Empowering Informed Decisions
As we've seen in the real-world applications section, solving equations empowers you to make informed decisions in your daily life. From budgeting and managing your finances to cooking and baking, planning home improvement projects, navigating travels, taking advantage of discounts, and even managing your time effectively, equations are a tool that helps you navigate the complexities of modern life.
By understanding how to set up and solve equations, you gain a greater sense of control over your circumstances. You're not just passively accepting whatever comes your way; you're actively analyzing situations, making calculations, and finding the best possible solutions. This sense of empowerment is a valuable asset in all aspects of life.
Equations: Building Confidence and Resilience
Finally, mastering equations builds confidence and resilience. When you successfully solve a challenging equation, you experience a sense of accomplishment that boosts your self-esteem. You realize that with effort and persistence, you can overcome obstacles and achieve your goals.
This confidence translates into a greater willingness to tackle new challenges and to persevere even when things get tough. The process of solving equations teaches you to break down complex problems, to think critically, and to learn from your mistakes. These are all essential skills for success in both your personal and professional life.
So, guys, keep practicing those equations! Whether you're solving for x in a math problem or solving for success in your life, the skills you develop along the way will serve you well. Remember, every equation you conquer is a step towards a brighter future. Keep challenging yourselves, stay curious, and never stop learning! You've got this!
