Solving Systems: X² + Y² = 9 And X² - Y² = 1

Hey guys! Today, we're diving into a fun little math problem: solving for x in a system of equations. We've got two equations here, and our goal is to find the values of x and y that satisfy both of them. It might seem a bit tricky at first, but trust me, we'll break it down step by step so it's super easy to understand. So, grab your pencils, and let's get started!

Understanding the Equations

Before we jump into solving, let's take a closer look at what these equations actually mean. The first equation, $x^2 + y^2 = 9$, should ring a bell for those familiar with geometry. It represents a circle! Think about it – the equation describes all the points (x, y) that are a fixed distance (the radius) away from the origin (0, 0). In this case, the radius is the square root of 9, which is 3. So, we're dealing with a circle centered at the origin with a radius of 3. Visualizing this circle can be incredibly helpful in understanding the possible solutions. Imagine drawing this circle on a graph; all the points that lie on this circle are potential solutions to our first equation. This geometric interpretation gives us a powerful visual aid, allowing us to think about the problem not just algebraically but also geometrically. By connecting the algebraic representation with a visual one, we can often gain deeper insights and a more intuitive understanding of the solutions we are looking for. It's like having two different lenses to view the same problem, each offering a unique perspective. The circle, with its smooth, continuous curve, represents an infinite set of points, each a potential solution. Our task is to find the specific points on this circle that also satisfy the second equation. This blend of algebraic manipulation and geometric intuition is what makes solving systems of equations so fascinating and rewarding. We are not just crunching numbers; we are exploring the relationship between equations and the shapes they represent, uncovering the hidden connections that lie beneath the surface of mathematical expressions.

Now, let's consider the second equation, $x^2 - y^2 = 1$. This one might look a little less familiar, but it actually represents a hyperbola. A hyperbola is a different kind of curve, with two separate branches that open away from each other. Unlike the circle, which is a closed curve, the hyperbola extends infinitely in both directions. The shape of the hyperbola is determined by the coefficients of the $x^2$ and $y^2$ terms, and the constant term on the right side of the equation. In our case, the hyperbola opens horizontally, since the $x^2$ term is positive. Just like with the circle, visualizing the hyperbola can be incredibly helpful. Imagine drawing this hyperbola on the same graph as the circle. The points where the circle and the hyperbola intersect are the solutions to our system of equations. These intersection points are the pairs of (x, y) values that satisfy both equations simultaneously. Thinking about the shapes of the curves helps us anticipate the number of solutions we might find. A circle and a hyperbola can intersect at most four times, so we know there are at most four solutions to our system. This geometric intuition provides a valuable check on our algebraic work, helping us to verify that our solutions are reasonable. The hyperbola, with its distinct shape and infinite extent, adds another layer of complexity to our problem. It is not as immediately intuitive as the circle, but its unique properties offer a fascinating contrast. The interaction between the circle and the hyperbola, their potential points of intersection, and the algebraic solutions they represent, all contribute to the richness and depth of this mathematical problem. By understanding the geometric nature of these equations, we gain a more complete and nuanced understanding of the solutions we seek.

Solving the System of Equations

Okay, so we know what our equations represent geometrically. Now, let's get down to the nitty-gritty of actually solving them. There are a couple of ways we can tackle this, but one of the most straightforward methods is using elimination. The elimination method is a powerful technique for solving systems of equations, particularly when the equations are structured in a way that allows for easy cancellation of terms. In our case, we have the equations $x^2 + y^2 = 9$ and $x^2 - y^2 = 1$. Notice that the $y^2$ terms have opposite signs. This is a golden opportunity for elimination! By simply adding the two equations together, the $y^2$ terms will cancel each other out, leaving us with an equation in terms of x only. This is a significant simplification, as it allows us to isolate x and solve for its values. The beauty of the elimination method lies in its ability to reduce a system of equations into a simpler, more manageable form. It transforms a problem with two variables into a problem with just one, making it much easier to solve. The key is to identify terms that can be eliminated, either directly or after some manipulation of the equations. In our case, the opposite signs of the $y^2$ terms make the elimination process particularly clean and efficient. But the method can be applied in a variety of situations, often involving multiplying one or both equations by a constant to create matching coefficients with opposite signs. This strategic manipulation allows us to eliminate any variable we choose, depending on the specific structure of the system. The elimination method is not just a mechanical procedure; it is a tool that requires careful observation and strategic thinking. It encourages us to look for patterns and relationships within the equations, and to devise a plan for simplifying the problem. By mastering the elimination method, we gain a powerful technique for tackling a wide range of systems of equations, from simple linear systems to more complex nonlinear ones. It is a fundamental tool in the mathematician's toolbox, and one that will serve us well in many different contexts.

So, let's add the equations together:

$(x^2 + y^2) + (x^2 - y^2) = 9 + 1$

This simplifies to:

$2x^2 = 10$

Now, we can solve for $x^2$ by dividing both sides by 2:

$x^2 = 5$

To find x, we take the square root of both sides:

x = old{\pm \sqrt{5}}

Awesome! We've found two possible values for x: $\sqrt{5}$ and $-\sqrt{5}$. But remember, we're not done yet. We still need to find the corresponding values for y.

To find the corresponding values for y, we'll substitute each value of x back into one of our original equations. It doesn't matter which equation we choose; we should get the same values for y in either case. Let's use the first equation, $x^2 + y^2 = 9$, because it looks a little simpler. Substituting the values back into an equation is a crucial step in solving systems of equations. It allows us to determine the values of the remaining variables, ensuring that our solutions satisfy all the equations in the system. This process of substitution is not just a mechanical one; it is a way of checking our work and ensuring the consistency of our results. By plugging the values we found for x back into the original equations, we can verify that they indeed lead to valid solutions for y. If we were to make a mistake in our earlier calculations, the substitution step would likely reveal the error, as the resulting equation would not hold true. Moreover, the choice of which equation to substitute into can sometimes affect the ease of the calculation. In our case, the first equation, $x^2 + y^2 = 9$, appears slightly simpler due to the absence of a negative sign. However, substituting into either equation should ultimately yield the same results for y. This flexibility in the substitution process is a testament to the power and robustness of algebraic methods. It allows us to approach the problem from different angles and to choose the path that seems most convenient. The substitution step is also a valuable opportunity to reinforce our understanding of the relationships between the variables in the system. By observing how the values of x and y interact within the equations, we can gain a deeper appreciation for the underlying mathematical structure. It is a process of not just finding solutions, but also of building intuition and developing a more holistic understanding of the problem.

First, let's substitute $x = \sqrt{5}$:

$(\sqrt{5})^2 + y^2 = 9$

This simplifies to:

$5 + y^2 = 9$

Subtracting 5 from both sides, we get:

$y^2 = 4$

Taking the square root of both sides:

$y = \pm 2$

So, when $x = \sqrt{5}$, we have two possible values for y: 2 and -2. This gives us two solutions: $(\sqrt{5}, 2)$ and $(\sqrt{5}, -2)$.

Now, let's substitute $x = -\sqrt{5}$:

$(-\sqrt{5})^2 + y^2 = 9$

Notice that $(-\sqrt{5})^2$ is also equal to 5, so we get the same equation as before:

$5 + y^2 = 9$

And, just like before, this leads to:

$y^2 = 4$

$y = \pm 2$

So, when $x = -\sqrt{5}$, we also have two possible values for y: 2 and -2. This gives us two more solutions: $(-\sqrt{5}, 2)$ and $(-\sqrt{5}, -2)$.

Identifying the Solutions

Alright, we've done the hard work and found our solutions! Now, let's compare our solutions to the options provided and see which ones match. We found four solutions in total: $(\sqrt{5}, 2)$, $(\sqrt{5}, -2)$, $(-\sqrt{5}, 2)$, and $(-\sqrt{5}, -2)$. These are the pairs of (x, y) values that satisfy both of our original equations simultaneously. Each of these points represents an intersection of the circle and the hyperbola we discussed earlier. The process of comparing our solutions to the given options is a crucial step in ensuring that we have correctly answered the question. It is a way of verifying that our algebraic work has led us to the solutions that are specifically requested. Moreover, it is an opportunity to reinforce our understanding of what it means for a point to be a solution to a system of equations. Each solution pair represents a point in the coordinate plane that lies on both the circle and the hyperbola. This geometric interpretation provides a valuable check on our algebraic results. If a given option does not match any of our calculated solutions, it indicates that it does not satisfy both equations simultaneously, and therefore is not a valid solution to the system. The comparison step also highlights the importance of careful attention to detail. We must ensure that the signs and values of the coordinates match exactly. A seemingly small difference, such as a sign error, can render an entire solution incorrect. By meticulously comparing our solutions to the options provided, we minimize the risk of overlooking subtle errors and maximize our chances of selecting the correct answers. This process of verification and validation is an essential component of mathematical problem-solving, ensuring the accuracy and reliability of our results. It is not simply about finding solutions; it is about confirming that the solutions we find are indeed the correct ones.

Looking at the options:

• $(-\sqrt{5}, -2)$ matches one of our solutions.
• $(\sqrt{5}, -2)$ matches one of our solutions.
• $(\sqrt{5}, 2)$ matches one of our solutions.
• $(-\sqrt{5}, 2)$ matches one of our solutions.

So, all the options provided are indeed solutions to the system of equations! We nailed it!

Conclusion

Solving systems of equations can seem daunting at first, but with a systematic approach and a little bit of practice, it becomes much easier. We started by understanding the geometric representation of our equations, then used the elimination method to solve for x, and finally substituted those values back in to find y. And boom! We found all the solutions. Remember, guys, math is like a puzzle – each piece fits together to create a beautiful solution. Keep practicing, and you'll be solving even the trickiest problems in no time!