Solve 12/x = 7/(x+5): Find The Unknown Variable
Hey guys! Today, we're diving into an exciting mathematical problem where we need to find the value of an unknown variable. Our equation is 12/x = 7/(x+5). This might look a bit intimidating at first glance, but don't worry, we'll break it down step-by-step and make it super easy to understand. So, grab your thinking caps, and let's get started!
Understanding the Equation
Before we jump into solving, let's make sure we understand what the equation is telling us. We have two fractions, 12/x and 7/(x+5), and they are equal to each other. The 'x' in our equation is the unknown that we need to find. Remember, in mathematics, we often use letters like 'x', 'y', or 'z' to represent values that we don't know yet. Our goal is to isolate 'x' on one side of the equation so we can figure out its value. This is a classic algebraic problem, and the techniques we'll use here are fundamental to solving many other types of equations. Think of it like this: we're trying to balance a scale. Whatever we do to one side of the equation, we must also do to the other side to keep the balance. This principle will guide us as we work through the steps.
Now, let's talk about why these kinds of problems are so important. Equations like this don't just live in textbooks; they pop up in all sorts of real-world situations. Imagine you're figuring out how long it will take to travel a certain distance at a specific speed, or you're calculating how to divide resources fairly among a group of people. These scenarios often involve relationships that can be expressed as equations, and being able to solve for unknowns is a crucial skill. So, as we tackle this problem, remember that we're not just learning math for the sake of it; we're developing a powerful tool that can help us make sense of the world around us.
Setting the Stage for Solving
Before we start crunching numbers, it's helpful to have a plan. Our plan of action for solving this equation is to first get rid of the fractions. Fractions can sometimes make equations look more complicated than they actually are, so eliminating them is often a good first step. To do this, we'll use a technique called cross-multiplication. Cross-multiplication is a handy shortcut that works when you have two fractions equal to each other. It's based on the principle that if a/b = c/d, then ad = bc. In other words, we multiply the numerator of the first fraction by the denominator of the second fraction, and vice versa. This will give us a new equation without any fractions.
Once we've eliminated the fractions, we'll have a more straightforward equation to work with. Our next step will be to simplify the equation by combining like terms. This might involve distributing numbers across parentheses or adding or subtracting terms on both sides of the equation. The goal here is to make the equation as clean and simple as possible. After simplifying, we'll be in a good position to isolate 'x'. This usually involves performing inverse operations to undo any operations that are being applied to 'x'. For example, if 'x' is being multiplied by a number, we'll divide both sides of the equation by that number. If a number is being added to 'x', we'll subtract that number from both sides. By carefully applying these steps, we'll gradually chip away at the equation until we have 'x' all by itself on one side. Finally, once we've found a value for 'x', it's always a good idea to check our answer. We can do this by plugging the value back into the original equation and making sure that both sides are equal. This helps us catch any mistakes we might have made along the way and gives us confidence that our solution is correct.
Step-by-Step Solution
Okay, let's get down to the nitty-gritty and solve this equation step-by-step. Remember, our equation is 12/x = 7/(x+5). The first thing we're going to do, as we discussed, is to get rid of those fractions using cross-multiplication. This means we'll multiply 12 by (x+5) and set that equal to 7 multiplied by x. So, we have:
12 * (x + 5) = 7 * x
Now, we need to simplify this equation. On the left side, we have 12 multiplied by the expression (x+5). To get rid of the parentheses, we'll use the distributive property. This means we'll multiply 12 by both x and 5. So, 12 times x is 12x, and 12 times 5 is 60. Our equation now looks like this:
12x + 60 = 7x
Great! We've eliminated the parentheses and made the equation a bit simpler. The next step is 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. To do this, we can subtract 7x from both sides of the equation. This will move the 7x term from the right side to the left side. Remember, whatever we do to one side, we must do to the other to keep the equation balanced. So, subtracting 7x from both sides, we get:
12x + 60 - 7x = 7x - 7x
Simplifying this, we have:
5x + 60 = 0
We're getting closer! Now, we want to isolate the term with 'x' (which is 5x) on one side of the equation. To do this, we can subtract 60 from both sides. This will move the 60 from the left side to the right side. So, subtracting 60 from both sides, we get:
5x + 60 - 60 = 0 - 60
Simplifying, we have:
5x = -60
Almost there! The last step is to isolate 'x' completely. Right now, 'x' is being multiplied by 5. To undo this multiplication, we'll divide both sides of the equation by 5. So, dividing both sides by 5, we get:
5x / 5 = -60 / 5
Simplifying, we finally have:
x = -12
Woo-hoo! We've found the value of 'x'. It's -12. But remember, we're not done yet. We need to check our answer to make sure it's correct.
Checking Our Solution
Checking our solution is a crucial step in any math problem. It's like proofreading a paper before you submit it – it helps you catch any errors and ensures that your answer is correct. To check our solution, we'll plug the value we found for 'x' (which is -12) back into the original equation. Our original equation was:
12/x = 7/(x+5)
Now, let's substitute -12 for 'x' in this equation. We get:
12/(-12) = 7/(-12 + 5)
Let's simplify both sides of the equation. On the left side, 12 divided by -12 is -1. So, we have:
-1 = 7/(-12 + 5)
On the right side, we need to simplify the denominator first. -12 plus 5 is -7. So, we have:
-1 = 7/(-7)
Now, 7 divided by -7 is also -1. So, the right side simplifies to:
-1 = -1
And there you have it! Both sides of the equation are equal. This means that our solution, x = -12, is correct. We've successfully solved the equation and verified our answer. Give yourself a pat on the back!
Real-World Applications
Now that we've conquered this equation, let's take a moment to appreciate why these skills are so valuable. Solving equations like 12/x = 7/(x+5) isn't just an academic exercise; it's a tool that can be applied to various real-world situations. Think about scenarios where you need to share resources proportionally, calculate rates and distances, or even design structures. These problems often involve relationships that can be expressed as equations, and your ability to solve for unknowns is key to finding the answers.
Imagine you're a chef scaling up a recipe for a large gathering. The recipe calls for a certain ratio of ingredients, and you need to figure out how much of each ingredient you'll need for the larger batch. This involves setting up proportions, which are essentially equations with fractions. Or, picture yourself planning a road trip. You know the distance you want to travel and the speed you'll be driving, and you need to calculate how long the trip will take. This involves using the formula distance = rate * time, which can be rearranged into an equation where you solve for the unknown time. These are just a couple of examples, but the possibilities are endless.
The beauty of algebra is that it provides a framework for thinking about and solving problems in a systematic way. By learning how to manipulate equations, you're developing a powerful problem-solving muscle that can be applied to all sorts of challenges, both inside and outside the classroom. So, keep practicing, keep exploring, and keep asking questions. The more you engage with math, the more you'll discover its relevance and its power to help you make sense of the world.
Practice Problems
To solidify your understanding, here are a few practice problems similar to the one we just solved. Try tackling these on your own, using the steps we discussed. Remember, the key is to break the problem down into smaller steps, stay organized, and check your answers. Don't be afraid to make mistakes – that's how we learn! If you get stuck, review the steps we went through together, or ask a friend or teacher for help.
- Solve for x: 5/x = 3/(x-2)
- Solve for y: 8/(y+1) = 4/y
- Solve for z: 10/z = 2/(z+3)
The more you practice, the more confident you'll become in your ability to solve equations. And remember, math is like any other skill – it gets easier with practice. So, keep at it, and you'll be amazed at what you can achieve!
Conclusion
So there you have it, guys! We've successfully solved the equation 12/x = 7/(x+5) and learned some valuable skills along the way. We started by understanding the equation and planning our approach. Then, we used cross-multiplication to eliminate fractions, simplified the equation by combining like terms, and isolated 'x' to find its value. Finally, we checked our solution to make sure it was correct. We also explored some real-world applications of these skills and practiced with some similar problems.
Remember, solving equations is a fundamental skill in mathematics and has wide-ranging applications in everyday life. By mastering these techniques, you're not just learning math; you're developing critical thinking and problem-solving abilities that will serve you well in many areas of your life. So, keep practicing, keep exploring, and never stop learning! And if you ever get stuck, don't hesitate to ask for help. There are plenty of resources available, from teachers and tutors to online tutorials and study groups. With perseverance and the right tools, you can conquer any mathematical challenge. Keep up the great work!
