Infinite Solutions: Understanding The Equation System

Have you ever encountered a system of equations that seems to have endless answers? It might sound puzzling, but it's a fascinating concept in mathematics! Let's dive into a specific example and explore why a system of equations can have infinite solutions. We'll break down the equation system:

$$\begin{aligned} -10 x^2-10 y^2 & =-300 \\ 5 x^2+5 y^2 & =150 \end{aligned}$$

and discuss the reasons behind its infinite solutions. Understanding these concepts not only helps in solving mathematical problems but also sharpens our analytical thinking. So, let's embark on this mathematical journey together!

Decoding the Equations: What Do They Represent?

Before we jump into why this system has infinite solutions, let's take a closer look at the equations themselves. Understanding the nature of these equations is the first step in solving the puzzle. Our equations are:

$$\begin{aligned} -10 x^2-10 y^2 & =-300 \\ 5 x^2+5 y^2 & =150 \end{aligned}$$

Do these equations look familiar? They might seem a bit intimidating at first, but let's simplify them. Notice that both equations involve ${x^2}$ and ${y^2}$ terms, and they are added together. This is a big clue! Equations of this form often represent circles. To confirm this, let's manipulate the equations into a more recognizable form. We can divide the first equation by -10 and the second equation by 5. This gives us:

$$\begin{aligned} x^2 + y^2 &= 30 \\ x^2 + y^2 &= 30 \end{aligned}$$

Now, do you see it? Both equations are in the standard form of a circle's equation: ${x^2 + y^2 = r^2}$, where ${r}$ is the radius of the circle. In our case, ${r^2 = 30}$, so both equations represent circles centered at the origin (0, 0) with a radius of ${\sqrt{30}}$. Recognizing this geometrical representation is key to understanding the solution set. Imagine drawing these circles on a graph. What do you think will happen? Will they intersect? Will they be separate? This visual representation will give us valuable insight into the nature of the solutions.

Now, let's consider what it means for a system of equations to have a solution. A solution is a point (x, y) that satisfies both equations simultaneously. Graphically, this means the points where the graphs of the equations intersect. If the graphs never intersect, there are no solutions. If they intersect at one or more discrete points, there are a finite number of solutions. But what if the graphs coincide completely? That's where the magic of infinite solutions comes in! We'll explore this concept further in the next section.

Unveiling Infinite Solutions: Identical Circles

So, we've established that both equations in our system represent circles centered at the origin with the same radius, ${\sqrt{30}}$. This is a crucial piece of the puzzle. Visualizing these circles is incredibly helpful. Imagine drawing them on the coordinate plane. What do you see? You'll notice that the two circles are exactly the same! They completely overlap. This is the key to understanding why we have infinite solutions.

Think about it: if two equations represent the same curve (in this case, a circle), every single point on that curve satisfies both equations. There's no single intersection point; instead, every point on the circle is a solution. This is the essence of infinite solutions. It's not just a few points; it's a continuum of points that make both equations true. This contrasts sharply with systems that have a finite number of solutions, where the graphs intersect at distinct points, or systems with no solutions, where the graphs never intersect.

Let's consider another way to look at this. Algebraically, we simplified the original equations to:

$$\begin{aligned} x^2 + y^2 &= 30 \\ x^2 + y^2 &= 30 \end{aligned}$$

Notice that the two equations are identical. This means that any pair of values (x, y) that satisfies one equation will automatically satisfy the other. There's no additional constraint or condition imposed by the second equation. The equations are redundant; they provide the same information. This redundancy is a telltale sign of infinite solutions. When solving a system of equations, we typically aim to find a unique solution, a specific set of values that satisfy all equations. However, in this case, the equations are not independent; they are dependent. This dependence leads to the infinite solution set.

The Answer Explained: Why Infinite Solutions Occur

Now that we've explored the nature of the equations and visualized the circles, let's directly address the question: Why does this system have infinite solutions? The answer lies in the fundamental relationship between the two equations. As we've seen, both equations represent the same circle. They are, in essence, different forms of the same equation. This is the core reason for the infinite solutions.

Let's consider the options provided in the original problem (which are not included here but are implied): Option A typically states that the equations represent parabolas that do not intersect. This is incorrect because our equations represent circles, not parabolas. Also, the circles do intersect – in fact, they overlap completely. The correct answer will highlight the fact that the equations represent the same circle.

To further solidify this understanding, let's think about what happens when we try to solve this system algebraically. If we use a method like substitution or elimination, we'll quickly find that one equation simply cancels out the other. For example, if we multiply the second equation by -2, we get:

$$-2(5x^2 + 5y^2) = -2(150)$$

$$-10x^2 - 10y^2 = -300$$

This is exactly the same as the first equation! When we try to eliminate one of the variables, we end up with an identity (e.g., 0 = 0), which is always true. This confirms that the equations are dependent and that there are infinite solutions. The solutions consist of all the points (x, y) that lie on the circle ${x^2 + y^2 = 30}$. There are infinitely many such points, as a circle is a continuous curve.

Beyond Circles: The Concept of Infinite Solutions in General

While our example focuses on circles, the concept of infinite solutions extends to other types of equations and systems. Understanding this broader perspective is crucial for mathematical proficiency. In general, a system of equations has infinite solutions when the equations are dependent, meaning one equation can be derived from the others. This dependence can manifest in various ways, not just through identical equations.

For instance, consider a system of linear equations:

$$\begin{aligned} x + y &= 5 \\ 2x + 2y &= 10 \end{aligned}$$

Notice that the second equation is simply twice the first equation. These lines are coincident; they overlap completely. Therefore, any point on the line ${x + y = 5}$ is a solution to the system, and there are infinitely many such points. The same principle applies to other types of equations, such as planes in three-dimensional space. If two equations represent the same plane, the system will have infinite solutions.

The key takeaway is that infinite solutions arise when the equations in a system provide redundant information. They don't impose independent constraints on the variables. This leads to a solution set that is not a finite set of points but rather a continuous curve, surface, or higher-dimensional object. Recognizing this dependence is essential for solving systems of equations and understanding their solution behavior.

In conclusion, the system of equations

$$\begin{aligned} -10 x^2-10 y^2 & =-300 \\ 5 x^2+5 y^2 & =150 \end{aligned}$$

has infinite solutions because both equations represent the same circle. This means every point on the circle satisfies both equations, leading to an infinite solution set. This concept of infinite solutions extends beyond circles and applies to any system where equations are dependent and provide redundant information. Exploring these mathematical concepts helps in developing critical thinking and problem-solving skills that are valuable in various aspects of life. For further exploration of systems of equations, consider visiting a trusted resource like Khan Academy's Systems of Equations Section.