Solving The Equation: $4=\frac{k+1}{3}+\frac{k+5}{5}$

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Hey guys! Today, we're diving into a fun little math problem where we need to solve for the variable k in the equation $4=\frac{k+1}{3}+\frac{k+5}{5}$. Don't worry, it might look intimidating at first, but we'll break it down step by step, making it super easy to understand. So, grab your pencils and let's get started!

Understanding the Equation

Before we jump into solving, let's take a moment to understand what the equation is telling us. We have a variable, k, mixed in with fractions and some basic arithmetic. Our goal is to isolate k on one side of the equation so we can find its value. This involves a series of algebraic manipulations, which, trust me, are simpler than they sound. We'll be using some key principles like finding a common denominator to combine fractions and then using inverse operations to get k all by itself. Think of it as a puzzle where we carefully move pieces around until we reveal the hidden answer. The beauty of algebra is that it gives us a systematic way to solve these kinds of problems, and once you get the hang of the basic techniques, you'll be able to tackle all sorts of equations with confidence. Remember, each step we take is designed to simplify the equation, bringing us closer to the solution.

Initial Assessment of the Equation

Okay, so let's start by really looking at our equation: $4=\frac{k+1}{3}+\frac{k+5}{5}$. The first thing that jumps out is those fractions. Fractions can sometimes make things look more complicated than they are, but don't fret! Our mission is to get rid of them as soon as we can. Notice that we have two fractions being added together on the right side of the equation. Both of these fractions have expressions involving k in their numerators (that's the top part of the fraction) and constant numbers in their denominators (the bottom part). This is a classic setup for a linear equation, meaning that when we solve for k, we're going to get a single numerical value as our answer. We're not dealing with any exponents or anything too crazy here, which is great news. Now, the presence of these fractions means that our first order of business will be to find a common denominator. This will allow us to combine the fractions into one single term, which will make our equation much easier to handle. We'll then be able to move things around, isolate k, and unveil the solution. So, with a little bit of fraction magic, we'll be well on our way to cracking this equation! Remember, every equation is just a puzzle waiting to be solved, and we've got all the tools we need to solve this one.

Why Finding a Common Denominator is Crucial

So, why are we making such a big deal about finding a common denominator? Well, the common denominator is absolutely crucial when you're dealing with adding or subtracting fractions. Think of it like this: you can't easily add apples and oranges unless you have a common unit, right? You might need to convert them both to "pieces of fruit" before you can say, "I have five pieces of fruit." The same logic applies to fractions. In our equation, $4=\frac{k+1}{3}+\frac{k+5}{5}$, we have one fraction with a denominator of 3 and another with a denominator of 5. To add these fractions together, we need to express them with the same denominator. This common denominator serves as our common unit, allowing us to combine the numerators (the top parts) of the fractions. Without a common denominator, it's like trying to add different slices of a pie when they're cut into different numbers of pieces – it just doesn't work! Finding the common denominator is a fundamental step in simplifying the equation. It transforms the fractions into a form that we can easily work with. It paves the way for us to combine terms, isolate the variable k, and ultimately find the solution. It's not just a mathematical trick; it's a necessary step for making the equation manageable and solvable. Once we have that common denominator in place, we'll be able to merge those fractions into a single, cohesive term, making the rest of the solution process flow much more smoothly. So, mastering this step is key to unlocking the solution to our equation and to tackling many other algebraic problems down the line.

Step-by-Step Solution

Okay, let's get down to business and solve this equation step-by-step. Remember, our goal is to isolate k, so we'll be using some algebraic techniques to make that happen.

Step 1: Finding the Common Denominator

The first thing we need to do, as we discussed, is to find a common denominator for the fractions $\frac{k+1}{3}$ and $\frac{k+5}{5}$. To do this, we look for the least common multiple (LCM) of the denominators, which are 3 and 5. The LCM of 3 and 5 is 15 because 15 is the smallest number that both 3 and 5 divide into evenly. Now, we need to rewrite each fraction with the denominator of 15. To do this, we multiply both the numerator and the denominator of each fraction by the factor that will turn the original denominator into 15.

For the first fraction, $\frac{k+1}{3}$, we need to multiply both the numerator and the denominator by 5, since 3 multiplied by 5 equals 15. This gives us:

$\frac{k+1}{3} \times \frac{5}{5} = \frac{5(k+1)}{15}$

For the second fraction, $\frac{k+5}{5}$, we need to multiply both the numerator and the denominator by 3, since 5 multiplied by 3 equals 15. This gives us:

$\frac{k+5}{5} \times \frac{3}{3} = \frac{3(k+5)}{15}$

Now, our original equation $4=\frac{k+1}{3}+\frac{k+5}{5}$ can be rewritten with the common denominator as:

$4 = \frac{5(k+1)}{15} + \frac{3(k+5)}{15}$

By finding the common denominator, we've taken a significant step towards simplifying the equation. We've transformed the fractions into a format where we can easily add them together. This sets us up nicely for the next step, where we'll combine the numerators and continue to isolate k. So, we've successfully navigated the first hurdle, and we're making good progress towards solving for k! Remember, every step we take brings us closer to the final answer, and mastering these fundamental techniques is key to tackling more complex problems in the future.

Step 2: Combining the Fractions

Alright, now that we've got our fractions sporting a common denominator, the next logical step is to combine them. Remember, we've transformed our equation to look like this: $4 = \frac{5(k+1)}{15} + \frac{3(k+5)}{15}$. Since both fractions now have the same denominator, we can go ahead and add their numerators. This is like saying if you have a quarter of a pizza and then you get another quarter, you now have two quarters – you're just adding the top numbers when the bottom numbers are the same.

To add the numerators, we simply add the expressions in the numerators together. So, we have $5(k+1) + 3(k+5)$. Before we can actually add these, we need to distribute the numbers outside the parentheses into the terms inside. This means we multiply 5 by both k and 1, and we multiply 3 by both k and 5. Let's do that:

• $5(k+1) = 5k + 5$
• $3(k+5) = 3k + 15$

Now we can add these together: $(5k + 5) + (3k + 15)$. To simplify this, we combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have the k terms (5k and 3k) and the constant terms (5 and 15). Adding these together gives us:

• $5k + 3k = 8k$
• $5 + 15 = 20$

So, when we add the numerators, we get $8k + 20$. Now we can rewrite our equation with this simplified numerator:

$4 = \frac{8k + 20}{15}$

We've made a lot of progress here! By finding the common denominator and combining the fractions, we've condensed the right side of our equation into a single fraction. This makes the equation much cleaner and easier to work with. Now, we're one step closer to isolating k and finding its value. We're on a roll, guys! Next up, we'll get rid of that fraction altogether, and you'll see how much simpler things become.

Step 3: Eliminating the Fraction

Okay, we've reached a major milestone – we've combined those fractions into a single, neat term. Our equation currently looks like this: $4 = \frac{8k + 20}{15}$. Now, let's tackle that fraction head-on and get rid of it! Fractions can sometimes feel like a hurdle, but there's a super straightforward way to eliminate them when they're in this form. The key is to use the magic of inverse operations.

Remember, the fraction bar is really just a fancy way of showing division. So, the right side of our equation is saying "(8k + 20) divided by 15." To undo this division, we need to do the opposite operation, which is multiplication. We're going to multiply both sides of the equation by 15. This is crucial – whatever we do to one side of the equation, we must do to the other to keep things balanced. It's like a seesaw; if you add weight to one side, you need to add the same weight to the other to keep it level.

So, let's multiply both sides of the equation by 15:

$15 \times 4 = 15 \times \frac{8k + 20}{15}$

On the left side, 15 times 4 is simply 60. On the right side, the multiplication by 15 cancels out the division by 15. It's like they shake hands and say goodbye, leaving us with just the numerator. This is exactly what we wanted!

So, after multiplying by 15, our equation becomes:

$60 = 8k + 20$

Look how much simpler that is! We've successfully eliminated the fraction, and the equation is now much more manageable. We've transformed it into a basic two-step equation, which is something we can easily solve. Getting rid of the fraction was a huge step forward, and we're now in the home stretch. We're getting closer and closer to unveiling the value of k. Next, we'll start isolating k by undoing the addition and then the multiplication.

Step 4: Isolating the Variable

Fantastic! We've ditched the fraction and our equation is looking super clean: $60 = 8k + 20$. Now it's time to really focus on isolating k. Remember, isolating a variable means getting it all by itself on one side of the equation. To do this, we'll need to undo any operations that are clinging to k. In our case, k is being multiplied by 8 and then having 20 added to it. We need to peel these layers away one by one, using the order of operations in reverse. Think of it like unwrapping a present; you have to undo the wrapping in the reverse order that it was put on.

The first thing we need to tackle is the addition of 20. To undo addition, we use subtraction. We're going to subtract 20 from both sides of the equation. Again, it's super important to do the same thing to both sides to keep the equation balanced:

$60 - 20 = 8k + 20 - 20$

On the left side, 60 minus 20 is 40. On the right side, the +20 and -20 cancel each other out, leaving us with just 8k. So, our equation now looks like this:

$40 = 8k$

We're almost there! We've managed to get the term with k all by itself on the right side. Now, there's just one more operation we need to undo: the multiplication by 8. To undo multiplication, we use division. We're going to divide both sides of the equation by 8:

$\frac{40}{8} = \frac{8k}{8}$

On the left side, 40 divided by 8 is 5. On the right side, the division by 8 cancels out the multiplication by 8, leaving us with just k. So, drumroll please… our equation now reads:

$5 = k$

Or, if we flip it around to the more traditional format:

$k = 5$

We did it! We've successfully isolated k and found its value. It took a few steps, but each one was a logical progression that brought us closer to the solution. Now, let's give ourselves a pat on the back and make sure our answer is correct by checking it.

Verifying the Solution

Okay, we've arrived at our solution: k = 5. But before we do a victory dance, it's always a good idea to double-check our work and make sure our answer is correct. This process is called verifying the solution, and it's a crucial step in problem-solving. Think of it as the final seal of approval on your mathematical masterpiece!

To verify our solution, we're going to take the value we found for k (which is 5) and plug it back into the original equation. If our solution is correct, then when we substitute 5 for k in the equation, both sides of the equation should be equal. If they're not, then we know we've made a mistake somewhere and need to go back and retrace our steps.

So, let's take our original equation: $4=\frac{k+1}{3}+\frac{k+5}{5}$, and substitute 5 for k:

$4 = \frac{5+1}{3} + \frac{5+5}{5}$

Now, we need to simplify both sides of the equation. Let's start with the right side. We'll perform the additions in the numerators first:

$4 = \frac{6}{3} + \frac{10}{5}$

Next, we'll simplify the fractions by dividing the numerators by the denominators:

$4 = 2 + 2$

Now, we just need to add 2 and 2 on the right side:

$4 = 4$

Ta-da! The left side of the equation (4) is equal to the right side of the equation (4). This means our solution, k = 5, is correct! We've successfully verified our answer, and we can confidently say that we've solved the equation.

Verifying the solution is like the icing on the cake. It gives you that extra bit of assurance that you've not only arrived at an answer but that it's the right answer. It's a great habit to get into, and it can save you from making mistakes in exams or in real-world problem-solving situations. So, always remember to verify your solutions – it's the mark of a true math whiz!

Conclusion

Alright, guys! We've reached the end of our mathematical journey, and what a journey it's been! We successfully solved the equation $4=\frac{k+1}{3}+\frac{k+5}{5}$, and we found that k equals 5. We didn't just stop there, though; we also took the extra step of verifying our solution to make sure we got it right. That's what I call a job well done!

Solving this equation involved a few key steps, each building on the previous one. We started by identifying the need for a common denominator to combine the fractions. We then found that common denominator (15) and rewrote the fractions accordingly. Next, we combined the fractions, which involved distributing, adding like terms, and simplifying. We then eliminated the fraction by multiplying both sides of the equation by the denominator. This left us with a simpler equation that we could solve by isolating k. We did this by undoing the addition and then the multiplication, finally arriving at our solution: k = 5. And, of course, we verified our solution by plugging it back into the original equation and confirming that both sides were equal.

This whole process highlights the power of algebra and the systematic way we can approach solving equations. It's not just about memorizing steps; it's about understanding the underlying principles and applying them logically. Each step we took was based on sound mathematical reasoning, and by following these steps, we were able to break down a seemingly complex problem into manageable parts. So, the next time you encounter an equation that looks intimidating, remember the techniques we used here: find common denominators, combine like terms, eliminate fractions, and isolate the variable. With practice and patience, you'll be solving equations like a pro in no time! Keep up the great work, everyone, and remember, math is just another puzzle waiting to be solved!