Solving Equations: Find Positive X And Check Your Work!

Hey math enthusiasts! Today, we're diving into the world of algebra to tackle a fun problem. Our goal? To solve algebraically for the positive value of x, with the condition that x is not equal to zero. And guess what? We'll even check our answer to make sure we've nailed it. Get ready to flex those problem-solving muscles! The equation we're working with is: $\frac{2 x+5}{7}=\frac{1}{x}$. This might look a little intimidating at first, but trust me, with a few simple steps, we'll crack this code.

First things first, let's understand what we're dealing with. We have a fraction on the left side of the equation and another fraction on the right. Our mission is to isolate x and find its value. Remember, x represents an unknown number, and our algebraic operations will help us reveal its identity. It's like a mathematical detective story, and we're the sleuths, using equations as clues. We need to be careful with negative and positive signs, paying attention to every detail in order to reach the correct answer. Alright, let's put on our thinking caps and begin. Are you ready?

So, how do we start? Well, the most common way to solve this is by using a little trick called cross-multiplication. This method helps us get rid of those pesky fractions and transforms our equation into something more manageable. Cross-multiplication involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice-versa. In our case, it means multiplying $(2x + 5)$ by x and multiplying $1$ by $7$. Let's write that down to keep track of our progress. Doing this gives us: $(2x + 5) * x = 1 * 7$. See, the fractions are gone! This is a much friendlier equation to work with. Now we are close to the solution of x, let's keep going. We'll simplify the expression, by multiplying all of the components. Are you excited to see the answer? I am.

Step-by-Step Solution: Unveiling the Value of x

Okay, guys, let's break down the solution step by step. We'll start with our cross-multiplied equation: $(2x + 5) * x = 7$. Expanding the left side, we get $2x^2 + 5x = 7$. Now, we want to set this equation to zero, which is a key step when solving quadratic equations. To do this, we subtract $7$ from both sides, giving us: $2x^2 + 5x - 7 = 0$. Voila! We now have a quadratic equation in standard form. Now, we are ready to find the value of x. Let's keep going.

This is where things get a bit more interesting. We have several options to solve a quadratic equation: factoring, completing the square, or using the quadratic formula. Let's start with factoring. Factoring involves finding two expressions that multiply together to give us our quadratic equation. In this case, we're looking for two numbers that multiply to give $-14$ (which is $2$ times $-7$) and add up to $5$. After some thinking, we find that the numbers are $7$ and $-2$. Using this, we can rewrite the equation as: $2x^2 + 7x - 2x - 7 = 0$. Now we can factor by grouping.

We group the first two terms and the last two terms: $x(2x + 7) - 1(2x + 7) = 0$. This gives us $(2x + 7)(x - 1) = 0$. For the product of two factors to equal zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for x. This gives us two possible solutions: $2x + 7 = 0$ or $x - 1 = 0$. Solving for x in the first equation, we get x = -rac{7}{2}. Solving for x in the second equation, we get $x = 1$. Remember, we're looking for the positive value of x. Therefore, our solution is $x = 1$.

Checking Your Answer: The Moment of Truth

Alright, folks, we've found our answer, but we're not done yet! In mathematics, it's always a good practice to check your answer. This ensures we haven't made any mistakes along the way and that our solution is correct. We'll substitute our value of x back into the original equation and see if it holds true. Remember our original equation: $\frac{2 x+5}{7}=\frac{1}{x}$. We found that $x = 1$. Let's plug it in.

Substituting $x = 1$ into the equation, we get: $\frac{2(1) + 5}{7}=\frac{1}{1}$. Simplifying the left side, we have $\frac{2 + 5}{7}=\frac{7}{7}$. The right side simplifies to $1$. So, we have $\frac{7}{7}=1$. And, of course, $\frac{7}{7}$ simplifies to $1$. This gives us $1 = 1$. Success! Our equation holds true. This means that our solution, $x = 1$, is indeed correct. We've not only solved the equation but also verified our answer. It's always a great feeling to know you've got the right answer, right? It's like finding a treasure. It’s important to check your work, as this improves your understanding and problem-solving skills.

Now, let's take a look at the alternative value of x: x = -rac{7}{2}. We were asked to find the positive value of x, so this solution is not valid. But just for fun, let's substitute this value into the equation and verify it. By substituting x = -rac{7}{2} into the original equation, we obtain: \frac{2(-rac{7}{2})+5}{7}=\frac{1}{-rac{7}{2}}. This simplifies to \frac{-7+5}{7}=-rac{2}{7}, or \frac{-2}{7}=-rac{2}{7}. This verifies our solution of x = -rac{7}{2}, but since we were looking for a positive value, we can safely disregard it.

Conclusion: You Did It!

And there you have it, guys! We've successfully solved for the positive value of x in the equation $\frac{2 x+5}{7}=\frac{1}{x}$. We used cross-multiplication to simplify the equation, rearranged it into a quadratic equation, and then factored it to find our potential solutions. Remember, it's a good practice to check your answer. By substituting the value back into the original equation, we were able to confirm that $x = 1$ is indeed the correct solution. Remember that solving equations is like a puzzle, requiring careful attention to detail and a systematic approach. With practice, you'll become more confident and proficient at solving even the most complex equations. The more problems you solve, the better you'll become at recognizing patterns and applying the correct methods.

Keep practicing, keep learning, and don't be afraid to challenge yourself. Mathematics is a beautiful and rewarding field, and every problem you solve is a victory. Keep up the great work, and remember to always double-check your answers. The more you work on these problems, the easier they become. Stay curious, stay persistent, and keep exploring the amazing world of mathematics! I hope you guys enjoyed today's session. Keep an eye out for more math adventures, and until next time, happy solving!