Solve $y + 2/y = 11/y$ For $y$

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Hey mathletes! Today we're diving into a super common type of algebra problem: solving for a variable, specifically $y$, in an equation. We've got the equation $y + \frac{2}{y} = \frac{11}{y}$, and our mission, should we choose to accept it, is to find the value(s) of $y$ that make this equation true. This isn't just about crunching numbers, guys; it's about understanding how to manipulate equations to isolate the variable we're interested in. We'll break down each step so you can follow along, and by the end, you'll be a pro at tackling equations like this. Let's get started!

Understanding the Equation

Alright, let's take a good look at our equation: $y + \frac{2}{y} = \frac{11}{y}$. What's the first thing that jumps out at you? Probably those fractions with $y$ in the denominator, right? This is a key feature we need to deal with. When you see variables in the denominator of fractions in an equation, it usually means you have to be a bit careful. The main thing to watch out for is the possibility of division by zero. In our equation, $y$ cannot be zero, because dividing by zero is a big no-no in math. So, right from the get-go, we're establishing that $y \neq 0$. This is a crucial condition that we'll need to keep in mind as we find our solutions. If we end up with $y=0$ as a potential answer, we'll have to discard it. This equation involves rational expressions, and working with them often involves clearing the denominators to simplify things. That's going to be our primary strategy here. We want to transform this equation into something a bit more familiar, like a linear or quadratic equation, which we know how to solve.

Clearing the Denominators: The First Step to Simplification

So, how do we get rid of those pesky denominators? The golden rule is to multiply every single term in the equation by the least common denominator (LCD). In this case, the only denominator we have is $y$. So, we're going to multiply both sides of the equation by $y$. Remember, we've already established that $y \neq 0$, so multiplying by $y$ is a valid operation that won't change the solutions to our equation (as long as we remember our condition $y \neq 0$). Let's write it out:

$$y \left( y + \frac{2}{y} \right) = y \left( \frac{11}{y} \right)$$

Now, we distribute the $y$ to each term on the left side:

$$y \cdot y + y \cdot \frac{2}{y} = y \cdot \frac{11}{y}$$

Let's simplify each part. The first term, $y \cdot y$, is just $y^2$. For the second term, $y \cdot \frac{2}{y}$, the $y$ in the numerator cancels out the $y$ in the denominator, leaving us with just $2$. And for the right side, $y \cdot \frac{11}{y}$, the $y$ in the numerator cancels out the $y$ in the denominator, leaving us with $11$. So, our equation now looks like this:

$$y^2 + 2 = 11$$

See how much simpler that looks? We've successfully eliminated the fractions! This is a huge win. This step is fundamental when dealing with rational equations; it transforms them into polynomial equations, which are generally much easier to solve. Always look for the LCD and multiply through. It's like performing a magic trick to make the complex parts disappear, leaving you with a clear path forward. This process is essential for simplifying the problem and setting us up for finding the actual values of $y$.

Isolating the Variable Term

Now that we have our simplified equation, $y^2 + 2 = 11$, our next goal is to get the term with the variable ($y^2$ in this case) all by itself on one side of the equation. Think of it like peeling an onion, layer by layer, until you get to the core. Right now, $y^2$ has a '+ 2' attached to it. To undo addition, we use subtraction. So, we're going to subtract 2 from both sides of the equation to maintain the balance:

$$y^2 + 2 - 2 = 11 - 2$$

On the left side, $+2$ and $-2$ cancel each other out, leaving us with just $y^2$. On the right side, $11 - 2$ equals $9$. So, our equation becomes:

$$y^2 = 9$$

We're getting closer, guys! We've successfully isolated the $y^2$ term. This is a critical intermediate step. It tells us that the square of our variable $y$ is equal to 9. This is a much more manageable form than the original equation. The key here is performing the inverse operation. Since we added 2 to isolate $y^2$, we subtracted 2. This careful manipulation is what allows us to simplify and eventually solve for $y$. It’s about applying the rules of algebra consistently to move terms around until the variable is isolated.

Solving for $y$: The Final Step

We've reached the point where $y^2 = 9$. Now, we need to find the value of $y$. If $y^2$ equals $9$, that means we are looking for a number that, when multiplied by itself, gives us $9$. This is where the concept of square roots comes in. To undo the squaring of $y$, we need to take the square root of both sides of the equation. It's super important to remember that when you take the square root of both sides of an equation like this, there are two possible answers: a positive one and a negative one. Why? Because both a positive number squared and its negative counterpart squared will result in a positive number. For example, $3 \times 3 = 9$, and also $(-3) \times (-3) = 9$.

So, we take the square root of both sides:

$$\sqrt{y^2} = \sqrt{9}$$

This gives us:

$$y = \pm 3$$

The symbol $\pm$ means 'plus or minus'. So, our two potential solutions are $y = 3$ and $y = -3$.

This is the final step in solving for $y$. We've applied the inverse operation of squaring, which is taking the square root. It's vital to recall that this operation yields two results, positive and negative, because squaring a negative number results in a positive number. This step directly gives us the values of $y$. We're almost done; we just need to quickly check our work, especially considering our initial condition.

Checking Our Solutions

We found two potential solutions: $y = 3$ and $y = -3$. Remember our very first condition? We said that $y$ cannot be zero because it appears in the denominator. Let's check if our solutions violate this. Both $3$ and $-3$ are definitely not zero, so we're good on that front! Now, let's plug each solution back into the original equation, $y + \frac{2}{y} = \frac{11}{y}$, to make sure they work.

Check for $y = 3$:

Substitute $y=3$ into the equation:

$$3 + \frac{2}{3} = \frac{11}{3}$$

To add the terms on the left, we need a common denominator, which is 3. So, $3$ can be written as $\frac{9}{3}$.

$$\frac{9}{3} + \frac{2}{3} = \frac{11}{3}$$

Adding the fractions on the left:

$$\frac{9+2}{3} = \frac{11}{3}$$

$$\frac{11}{3} = \frac{11}{3}$$

This is true! So, $y = 3$ is a valid solution.

Check for $y = -3$:

Substitute $y=-3$ into the equation:

$$-3 + \frac{2}{-3} = \frac{11}{-3}$$

Again, we need a common denominator. $-3$ can be written as $\frac{-9}{3}$.

$$\frac{-9}{3} + \frac{2}{-3} = \frac{11}{-3}$$

Be careful with the signs here. $\frac{2}{-3}$ is the same as $-\frac{2}{3}$, and $\frac{11}{-3}$ is the same as $-\frac{11}{3}$.

$$\frac{-9}{3} - \frac{2}{3} = \frac{-11}{3}$$

Now, combine the terms on the left:

$$\frac{-9-2}{3} = \frac{-11}{3}$$

$$\frac{-11}{3} = \frac{-11}{3}$$

This is also true! So, $y = -3$ is also a valid solution.

This checking step is super important, especially when dealing with equations that might introduce extraneous solutions (solutions that look good but don't actually work in the original equation). By plugging our answers back in, we confirm their validity. It's the final seal of approval on our mathematical detective work.

Conclusion: The Solutions

After carefully working through the steps – clearing the denominators, isolating the variable term, taking the square root, and checking our answers – we've determined that both $y = 3$ and $y = -3$ satisfy the original equation $y + \frac{2}{y} = \frac{11}{y}$. These are our two solutions. So, when asked to solve for $y$ and separate multiple solutions with commas, the answer would be $3, -3$. Nicely done, everyone!

This problem is a great example of how algebraic manipulation, combined with a keen eye for potential pitfalls like division by zero, leads us to the correct answer. Keep practicing these kinds of problems, and you'll find that solving equations becomes second nature. Remember, math is all about building on fundamental principles, and each solved problem adds another tool to your mathematical toolbox. Keep up the great work!