Solving Equations: A Step-by-Step Guide

Hey Plastik Magazine readers! Ever stumbled upon an equation that seems like a total brain-buster? Don't sweat it! Today, we're diving deep into the world of equations, specifically tackling how to find all real solutions for a tricky one: \frac{1}{x-1}+rac{1}{x+6}=rac{9}{8}. Sounds intimidating, right? But trust me, with the right approach, we can break it down into manageable chunks. This article is your ultimate guide, where we'll explore each step and make sure you understand every single move. We are going to go through a detailed process, guys, so let’s get started. Get ready to flex those math muscles and feel the satisfaction of solving it! This is gonna be a blast, and you'll be feeling super confident in no time! So, let's get down to business and figure out how to conquer these types of problems. By the end, you'll be a pro at solving similar equations. Keep in mind, solving equations is not just about finding answers; it's about developing critical thinking and problem-solving skills that are super valuable in all aspects of life. This is your chance to really shine and get comfortable with math! The equation $\frac{1}{x-1}+\frac{1}{x+6}=\frac{9}{8}$ can look tricky at first glance, but fear not! With a systematic approach, we can simplify and solve it. This is where we learn, develop, and grow! Are you ready to dive into the core of the problem? Let us begin!

Step 1: Clearing the Fractions

Alright, first things first, let's get rid of those pesky fractions. The goal here is to transform the equation into a more manageable form. We can do this by multiplying both sides of the equation by the least common denominator (LCD) of all the fractions involved. The denominators in our equation are $x-1$, $x+6$, and $8$. Therefore, the LCD is $8(x-1)(x+6)$. Now, here's the fun part – we'll multiply each term by the LCD. This might look a little messy at first, but stick with it, and you'll see how beautifully it simplifies. Multiply both sides by $8(x-1)(x+6)$: $8(x-1)(x+6) \cdot (\frac{1}{x-1} + \frac{1}{x+6}) = 8(x-1)(x+6) \cdot \frac{9}{8}$. Distribute the LCD on the left side: $8(x+6) + 8(x-1) = 9(x-1)(x+6)$. Expand the terms: $8x + 48 + 8x - 8 = 9(x^2 + 5x - 6)$. Simplify and combine the terms: $16x + 40 = 9x^2 + 45x - 54$. At this point, everything should be falling into place. It's really awesome to see it come together! Next up, we will rearrange this equation so that it equals zero, bringing us closer to solving for x. Remember, the key is to stay organized and keep track of each step. You've got this! We're essentially transforming the equation into a quadratic equation, which we can solve using various methods, such as factoring, completing the square, or the quadratic formula. Let’s do it!

Step 2: Simplifying into a Quadratic Equation

Now, let's move everything to one side to get our equation in the standard quadratic form, which is $ax^2 + bx + c = 0$. We've already done the heavy lifting, and this part is relatively straightforward. Subtract $16x$ and $40$ from both sides: $9x^2 + 45x - 54 - 16x - 40 = 0$. Simplify it: $9x^2 + 29x - 94 = 0$. Boom! We now have a standard quadratic equation. Now what? Well, our next step is to actually solve this equation for $x$. There are several methods we can use, so let’s get started. You can either factor, complete the square, or use the quadratic formula. For this equation, factoring might be a bit tricky, so let's use the quadratic formula. It’s a reliable way to solve any quadratic equation, regardless of how complex it looks. Ready to dig in? Here is the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Where $a$, $b$, and $c$ are the coefficients from our quadratic equation. It is also important to note the importance of staying organized. Writing down each step and double-checking your work can save you a ton of time and prevent silly mistakes. Just a friendly reminder to be careful and double-check your work, guys.

Step 3: Solving the Quadratic Equation

Alright, time to roll up our sleeves and apply the quadratic formula. Remember our equation: $9x^2 + 29x - 94 = 0$. We can identify our coefficients: $a = 9$, $b = 29$, and $c = -94$. Now, let's plug these values into the quadratic formula and find our solutions: $x = \frac{-29 \pm \sqrt{29^2 - 4 \cdot 9 \cdot (-94)}}{2 \cdot 9}$. Simplify under the square root: $x = \frac{-29 \pm \sqrt{841 + 3384}}{18}$. Calculate the values: $x = \frac{-29 \pm \sqrt{4225}}{18}$. Find the square root: $x = \frac{-29 \pm 65}{18}$. Now, we have two possible solutions, let's figure them out. First solution: $x_1 = \frac{-29 + 65}{18} = \frac{36}{18} = 2$. Second solution: $x_2 = \frac{-29 - 65}{18} = \frac{-94}{18} = -\frac{47}{9}$. Wow! We did it! Now we have our two potential solutions for x. But are we completely done? Nope! Before we celebrate, we must check these solutions.

Step 4: Checking the Solutions

Okay, before we declare victory, it's crucial to check if our solutions are valid. Sometimes, when solving equations with fractions, we might get solutions that don't actually work in the original equation. These are called extraneous solutions. So, we'll take our two potential solutions and plug them back into the original equation $\frac{1}{x-1} + \frac{1}{x+6} = \frac{9}{8}$. Let’s start with $x = 2$: $\frac{1}{2-1} + \frac{1}{2+6} = \frac{1}{1} + \frac{1}{8} = 1 + \frac{1}{8} = \frac{9}{8}$. This solution is valid! Excellent work! Now, let’s check $x = -\frac{47}{9}$. $\frac{1}{-\frac{47}{9}-1} + \frac{1}{-\frac{47}{9}+6} = \frac{1}{-\frac{56}{9}} + \frac{1}{\frac{7}{9}} = -\frac{9}{56} + \frac{9}{7} = -\frac{9}{56} + \frac{72}{56} = \frac{63}{56} = \frac{9}{8}$. This solution is also valid! Both of our solutions check out, meaning we didn't get any extraneous solutions this time around. That doesn't happen all the time, so remember to always check your answers! Now, we can give ourselves a pat on the back. We have successfully found the real solutions for our equation. So proud of you guys!

Conclusion

And there you have it, folks! We've successfully navigated the equation $\frac{1}{x-1} + \frac{1}{x+6} = \frac{9}{8}$ from start to finish. We cleared fractions, simplified into a quadratic equation, solved it using the quadratic formula, and checked our solutions to make sure they were valid. Remember, the key is to break down complex problems into smaller, manageable steps. Practice makes perfect, so keep solving equations, and you'll become a pro in no time. This is a journey, and every step counts. Always remember to double-check your work, stay organized, and never be afraid to ask for help. And that's all, folks! Hope you enjoyed the article. Stay tuned for more math adventures with Plastik Magazine! Now, go forth and conquer those equations! Feel free to leave a comment with any questions. We're here to help! Keep learning, keep growing, and keep shining! You are all amazing, and I am so proud of you guys!