Solving Equations: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of equations, specifically tackling how to solve for a variable. The equation we'll be working with is: $\frac{9}{15}=\frac{24 k}{40}$. Don't worry if this looks a little intimidating at first; we'll break it down into easy-to-understand steps. Solving equations is a fundamental skill in math, and once you get the hang of it, you'll be able to tackle more complex problems with confidence. So, grab your pencils and let's get started! This problem is a classic example of a proportion, and we can solve it using a few different methods. We'll explore the cross-multiplication method, which is a super efficient way to solve these kinds of problems. This approach is all about getting rid of those fractions and isolating the variable, which in this case is 'k'. Trust me; by the end of this, you'll feel like a pro at solving equations like this one. Remember, practice makes perfect, so don't be afraid to try more problems after we're done here. Let's make sure we understand the principles behind this type of problem, and how to apply them. Understanding these principles will make solving more complex problems less daunting. Keep an open mind, be patient with yourself, and let's unlock the secrets of equation solving!

Step-by-Step Solution: Cross-Multiplication

Alright, let's get down to the nitty-gritty and solve this equation step-by-step. The first method we'll use is cross-multiplication. This is a handy technique that simplifies the equation by eliminating the fractions. Here’s how it works: you multiply the numerator of the first fraction by the denominator of the second fraction, and set that equal to the product of the denominator of the first fraction and the numerator of the second fraction. In our equation $\frac{9}{15}=\frac{24 k}{40}$, we multiply 9 by 40 and set that equal to 15 times 24k. That gives us: 9 * 40 = 15 * 24k. Now, let’s do the multiplication. 9 times 40 equals 360, and 15 times 24k equals 360k. So, we now have a new equation: 360 = 360k. See how much simpler that looks? From here, the goal is to isolate 'k', which means we need to get 'k' all by itself on one side of the equation. To do this, we need to get rid of the 360 that's being multiplied by 'k'. What is the solution? What does the value of 'k' represent? Remember, the value of 'k' tells us the number that makes the original equation true. The number we find is the solution that satisfies the equation. The value of 'k' is what makes the left side equal the right side. The value of 'k' is a single number that holds the key to unlocking the true statement of this equation. This is the single value that confirms our original equation. By solving for 'k', we are essentially finding the 'balance' of the equation. Finding that balance is the heart of solving any equation. This balance is what makes an equation true, a statement where the left and right sides are the same. This is the ultimate goal when we are solving the equation.

Isolating the Variable

To isolate 'k', we need to divide both sides of the equation by 360. This is because we want to undo the multiplication. So, we divide both sides by 360: 360 / 360 = 360k / 360. When you divide 360 by 360, you get 1. And when you divide 360k by 360, you’re left with just 'k'. So, we have 1 = k, or k = 1. That's it! We've solved for 'k'. The value of k that makes the equation $\frac{9}{15}=\frac{24 k}{40}$ true is 1. We did it! This method is a fast and effective way to solve proportions, and you can apply it to a wide range of similar problems. Now that we have solved for 'k', what does this answer mean? The value of k = 1 means that if you substitute 1 for 'k' in the original equation, the equation will be balanced. It is this balance that we strive for when solving equations. Think about it: our original equation, when we plugged in 1 for k, would say that the left side equals the right side, so it makes it a valid statement. Therefore, solving this equation is not just about finding a number; it's about finding the number that maintains equality within the equation. It's about finding the balance! The joy of solving equations comes from seeing the relationship and understanding how each part of the equation connects. That is the beauty of mathematics. Remember, the journey of solving an equation is just as important as the answer itself. Understanding the steps allows you to solve a variety of equations, and each equation you solve makes the next one easier.

Simplifying the Fractions

Before we dive into cross-multiplication, we can simplify the fractions in the original equation. Simplification makes the numbers smaller and the equation easier to manage. Let's look at our equation again: $\frac{9}{15}=\frac{24 k}{40}$. Both 9 and 15 are divisible by 3. So, we can divide both the numerator and the denominator of the first fraction by 3. This gives us $\frac{3}{5}$. Now let's simplify $\frac{24 k}{40}$. Both 24 and 40 are divisible by 8. Dividing both the numerator and denominator by 8, we get $\frac{3k}{5}$. Our simplified equation becomes $\frac{3}{5}=\frac{3k}{5}$. Now, let's solve this. Notice that both fractions have the same denominator, 5. Since the denominators are the same, the numerators must be equal to each other for the fractions to be equal. That means 3 = 3k. Now, to isolate 'k', divide both sides by 3, so 3/3 = 3k/3, and we get 1 = k, or k=1. And there we have it, the same answer! We get the same value for 'k' whether we simplify the fractions first or not. Simplifying can sometimes make the numbers smaller and the calculations easier, but it’s not strictly necessary. Both methods are valid and will lead you to the correct answer. The key is to choose the method you feel most comfortable with, and the one you can perform most accurately. Either way, you arrive at the correct value for k, which, as we mentioned before, is 1. The value of k = 1 makes the equation valid, which means that the original equation is true. Both methods, simplified or non-simplified, show us how to arrive at the same solution. In our case, the value of k = 1 balances the equation.

The Importance of Simplifying

Simplifying fractions is a handy skill that makes calculations easier. By reducing fractions to their simplest form, you work with smaller numbers, which reduces the chance of making calculation errors. Simplifying can also make it easier to spot patterns in equations, or find different paths to the same answer. It is a time-saving technique that makes complex calculations less complex. In this case, simplifying allowed us to see more directly how the numerators related to each other. Keep in mind that not all fractions can be simplified. If the numerator and the denominator do not share a common factor other than 1, the fraction is already in its simplest form. When you simplify, you're not changing the value of the fraction; you're just expressing it in a more efficient way. For example, $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent, but $\frac{1}{2}$ is simpler. Mastering the art of simplification is like having a secret weapon in your math toolkit. It can save you time, reduce errors, and build your confidence. It helps build a strong foundation for tackling more complex math problems. Always remember that the ultimate goal is to solve the equation. Whether you choose to simplify first or not, just make sure you can accurately solve the problem. So, practice simplifying as part of your math journey.

Verifying the Solution

Once we've solved for 'k', it's always a good practice to verify our answer. This means we'll substitute the value we found for 'k' back into the original equation and check if the equation holds true. Our original equation was $\frac{9}{15}=\frac{24 k}{40}$, and we found that k = 1. Let's substitute 1 for 'k': $\frac{9}{15}=\frac{24 * 1}{40}$. Now, let's simplify the right side of the equation. 24 times 1 is 24, so we have $\frac{24}{40}$. We already know that $\frac{24}{40}$ can be simplified to $\frac{3}{5}$. Does this make the equation true? To check this, let's simplify the left side of the equation, $\frac{9}{15}$. Both 9 and 15 are divisible by 3. Dividing both the numerator and denominator by 3, we get $\frac{3}{5}$. So, our simplified equation is $\frac{3}{5}=\frac{3}{5}$. Since the left side of the equation equals the right side, our solution is correct. We have successfully verified that k = 1 is the correct solution to the equation $\frac{9}{15}=\frac{24 k}{40}$. Verifying your answer is a crucial step in problem-solving. It helps to catch any mistakes you might have made during the calculation, and it builds your confidence in your solution. This also ensures that we didn’t make any mistakes. This is a very important part of solving equations, because it makes sure that our solution is correct. This step is a check to make sure the answer is correct and valid. This check reassures us that we have correctly solved for 'k'. Verifying your answer will ensure that you correctly solve an equation. That feeling of getting the right answer is amazing! The confidence you gain is invaluable and will fuel your determination to tackle more complex mathematical challenges. That is the beauty of the learning process!

Conclusion

So, there you have it, guys! We've successfully solved the equation $\frac{9}{15}=\frac{24 k}{40}$ using cross-multiplication, and we've verified our answer. We also looked at how simplifying the fractions can make the equation easier to manage. Remember, the key to solving equations is to isolate the variable, and the best way to do that is to perform opposite operations on both sides of the equation. Also, always remember to verify your answer. This ensures that you have found the correct solution. Keep practicing these techniques, and you'll become a pro at solving equations in no time. Mathematics is like a game, and solving equations is just one of its many exciting levels. Keep exploring, keep practicing, and keep having fun! Remember that every problem you solve is a victory. The more equations you solve, the better you’ll get! Believe in yourself, and keep practicing; your mathematical skills will continue to improve. Now go out there and solve some equations! Keep up the great work; you will be successful! Happy solving, and keep up the great work! You've got this! We hope this guide has been helpful, and we'll see you in the next lesson!