Solving Equations: Find The Root Divisible By 9

Hey guys! Today, we're diving into some algebra fun! We're going to solve a couple of equations and then see if we can find a root that's divisible by 9. Sounds simple, right? Let's get started and break down each step so you can follow along easily. Trust me, it's not as scary as it looks – just a little bit of number crunching! We'll tackle two equations, and I'll walk you through how to isolate the variable, solve for it, and then check our answer. The goal is to get comfortable with these types of problems. We'll also look at the concept of divisibility, and how it helps us determine if a number is a multiple of another. So grab a pen and paper and let's get into the thick of it! Are you ready to become equation-solving pros? I know you can do it! Remember, the more problems you solve, the better you become. This is the perfect opportunity to sharpen your skills and get confident in your abilities. I'll try to make it as clear as possible. Each step is carefully considered, and I'll explain every trick along the way, ensuring you grasp the underlying principles of each calculation. So, let's see which of our roots is divisible by 9!

Equation 1: Unraveling the First Equation

Let's start with the first equation. Our main keyword here is solving the equation. Here is the equation we need to solve:

(76000 - 8 * x) : 4 = 17866

Our mission? To find the value of 'x'. The first step, guys, is to get rid of the division by 4. We can do this by multiplying both sides of the equation by 4. This is a super important step to follow. It isolates the expression containing 'x'. Remember, whatever you do to one side of the equation, you must do to the other to keep things balanced. This might seem like a small detail, but following this simple rule ensures we're always on the right track. Multiplying the right side by 4, we'll get 17866 * 4 = 71464. This gives us:

76000 - 8 * x = 71464

Now, our aim is to isolate the term with 'x'. This is our next focus. We'll do this by subtracting 76000 from both sides of the equation. This will leave us with the term containing x all by itself on one side. This method helps keep the equation balanced, ensuring our steps are mathematically sound. So, let's do the math. Subtracting 76000 from both sides, we get:

-8 * x = 71464 - 76000

-8 * x = -4536

Almost there! Now, to find 'x', we need to divide both sides of the equation by -8. This will give us the value of x. The crucial concept here is to remember that dividing by a negative number changes the sign. The goal is to isolate 'x'. This helps in arriving at the final value. Let's perform the division:

x = -4536 / -8

x = 567

Alright, we've found the value of 'x'! The answer is 567. Now, before we move on, let's take a quick pause. We have successfully solved the first equation. Good job! What a journey!

Checking for Divisibility by 9

Our second major task is to check if 567 is divisible by 9. So, how do we do this? The easiest way is to use the divisibility rule for 9. A number is divisible by 9 if the sum of its digits is divisible by 9. This is our focus now. So, let's add the digits of 567: 5 + 6 + 7 = 18. Since 18 is divisible by 9 (18 / 9 = 2), then 567 is also divisible by 9. Amazing, we found our answer!

Equation 2: Solving the Second Equation

Now, let's move on to the second equation. This is going to be so exciting! Our new keyword will be equation solving. Here's what we're dealing with:

(y : 17 + 419) * 9 = 4230

Our goal here is to find the value of 'y'. The first step is to undo the multiplication by 9. We can do this by dividing both sides of the equation by 9. This helps isolate the expression with 'y'. Always remember to keep things balanced by performing the same operation on both sides. Let's see:

(y : 17 + 419) = 4230 / 9

(y : 17 + 419) = 470

Next, we need to isolate the term containing 'y'. We'll do this by subtracting 419 from both sides. This helps simplify the expression with 'y'. Ensure all calculations are precise. This ensures our solution is correct. It's all about keeping things organized and methodical. Let's do that:

y : 17 = 470 - 419

y : 17 = 51

Now, to find 'y', we need to get rid of the division by 17. We do this by multiplying both sides of the equation by 17. This helps in finding the value of 'y'. Let's go ahead and do that. This method allows us to uncover the final value. Now, watch this:

y = 51 * 17

y = 867

Fantastic! We've found the value of 'y'! The answer is 867. That was fantastic. Now that we have our answer, let's double-check that it's correct.

Checking for Divisibility by 9

Here is the next crucial step: checking if 867 is divisible by 9. Remember, we can use the divisibility rule for 9. We simply add up the digits and see if the sum is divisible by 9. Let's calculate: 8 + 6 + 7 = 21. Hmm, 21 is not divisible by 9. (21 / 9 = 2.33). That means 867 is not divisible by 9. Thus, only x = 567 is divisible by 9. We did it, guys!

Conclusion: Final Thoughts

Alright, guys! We've successfully solved both equations and checked which of our roots is divisible by 9. Remember, the key to success in solving these problems is to stay organized and focused. Follow the steps carefully, keep track of each operation, and always check your work. Practice makes perfect. By tackling more problems, you'll become more confident in your abilities. The more you work on these problems, the better you'll get! So, keep practicing, stay curious, and enjoy the world of math! We hope you had fun. See you in the next lesson!