Infinite Solutions: Understanding Systems Of Equations

Let's dive into the fascinating world of systems of equations and uncover why certain systems boast an infinite number of solutions. We'll specifically analyze the system: \begin{aligned} -10x^2 - 10y^2 &= -300 \ 5x^2 + 5y^2 &= 150 \\end{aligned}. We'll explore why this particular system doesn't just have one or a few solutions, but rather an infinite number of them, and why the given options in your question are accurate.

Decoding Systems of Equations and Their Solutions

First off, let's break down what a system of equations actually is. Imagine you have two or more equations, each with variables (like x and y in our example). When we talk about a solution to this system, we're looking for values for those variables that make all the equations true simultaneously. Think of it as finding the common ground, the point(s) where all the equations agree. The kind of system of equations we are dealing with has infinite solutions, meaning that there isn't just one or two pairs of x and y that satisfy both equations, but rather a whole bunch of them!

Now, systems of equations can behave in a few different ways. They can have a single, unique solution (the equations intersect at one point). They might have no solution at all (the equations are parallel and never intersect). Or, as in our case, they can have an infinite number of solutions. This happens when the equations are essentially the same, just written in a different form. The graphs intersect at every point.

Let's analyze the given equations to understand this concept better and show the infinite solutions of the provided system of equations. Our goal is to determine why the given system has infinite solutions, relating to the behavior of their graphical representations.

The Heart of the Matter: Understanding the Equations

Our system presents two equations. The first, $-10x^2 - 10y^2 = -300$, might look a bit intimidating at first. Let's simplify it. We can divide both sides of the equation by -10, which gives us $x^2 + y^2 = 30$. This equation actually represents a circle centered at the origin (0, 0) with a radius of the square root of 30. The second equation is $5x^2 + 5y^2 = 150$. Similar to the first equation, let's simplify it by dividing both sides by 5. This simplifies to $x^2 + y^2 = 30$. This is, as you can see, the exact same equation as we derived from the first one. So, what initially seemed like two distinct equations are, in reality, two different representations of the same circle.

Because both equations represent the same circle, they intersect at every single point on that circle. Every point on the circle satisfies both equations, meaning there are infinitely many solutions.

So, why does this system have infinite solutions? Because the equations are not independent. They are dependent and represent the same geometric shape. This is a very common scenario when dealing with this kind of system. The graphical representation of this is obvious since they are the same circle, they overlap each other, leading to an infinite number of solutions.

Why the Other Options are Incorrect

Now, let's consider why the other answer choices are not correct. In the context of the answers given, the option is