Infinite Solutions: System Of Equations Explained

Have you ever stumbled upon a system of equations that seems to have an endless number of answers? It's a fascinating concept in mathematics, and in this article, we're going to dive deep into understanding why such systems exist. Specifically, we'll be looking at the system:

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

and exploring the reasons behind its infinite solutions. Let's embark on this mathematical journey together!

Understanding Systems of Equations

Before we tackle the specific system at hand, let's take a moment to understand what a system of equations is. At its core, a system of equations is a set of two or more equations that share variables. The solution to a system of equations is the set of values for the variables that satisfy all equations simultaneously. Graphically, the solution represents the point(s) where the graphs of the equations intersect.

When dealing with systems of equations, there are three possible scenarios:

1. Unique Solution: The system has one and only one solution. The graphs of the equations intersect at a single point.
2. No Solution: The system has no solution. The graphs of the equations do not intersect.
3. Infinite Solutions: The system has an infinite number of solutions. The graphs of the equations coincide, meaning they are essentially the same line or curve.

Our focus here is on the third scenario – infinite solutions. To understand why a system might have infinite solutions, we need to examine the relationships between the equations.

Identifying Infinite Solutions

Systems with infinite solutions often share a special characteristic: the equations are dependent. Dependent equations are essentially multiples of each other. This means that one equation can be obtained by multiplying the other equation by a constant. When graphed, dependent equations represent the same line or curve, leading to an infinite number of intersection points, and thus, infinite solutions.

Now, let's circle back to our system and analyze it in this context.

Analyzing the Given System of Equations

Let's revisit the system of equations we're investigating:

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

To determine if this system has infinite solutions, we need to check if the equations are dependent. One way to do this is to manipulate one equation and see if it can be transformed into the other.

Let's start with the second equation:

$$5x^2 + 5y^2 = 150$$

If we multiply this entire equation by -2, we get:

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

Simplifying, we have:

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

Notice anything familiar? This is exactly the first equation in our system! This crucial observation reveals that the two equations are indeed dependent. One equation is simply a multiple of the other.

The Geometric Interpretation: Circles

To further solidify our understanding, let's consider the geometric representation of these equations. Both equations are in the form of a circle equation. A general equation of a circle centered at the origin (0, 0) is:

$$x^2 + y^2 = r^2$$

where r is the radius of the circle. Let's rewrite our equations in this standard form.

For the first equation, $-10x^2 - 10y^2 = -300$, we can divide both sides by -10:

$$x^2 + y^2 = 30$$

For the second equation, $5x^2 + 5y^2 = 150$, we divide both sides by 5:

$$x^2 + y^2 = 30$$

As you can see, both equations simplify to the same equation: $x^2 + y^2 = 30$. This equation represents a circle centered at the origin with a radius of $\sqrt{30}$. Since both equations represent the same circle, they overlap completely. This overlap visually confirms the existence of infinite solutions – every point on the circle satisfies both equations.

Why Infinite Solutions Occur

Now, let's crystallize the reason behind the infinite solutions. The core principle is that dependent equations represent the same relationship between the variables. In our case, both equations describe the same circle. When two equations in a system represent the same curve, every point on that curve is a solution to the system. Since a circle has an infinite number of points, the system has an infinite number of solutions.

The equations are not providing independent pieces of information. Knowing one equation essentially tells you everything the other equation does. There's no unique constraint that narrows down the solution to a single point or a finite set of points.

The Pitfalls of Parabolas and Other Conics

The initial statement in the problem mentioned parabolas, which might have caused some confusion. While parabolas are conic sections, like circles, they behave differently in systems of equations. Two parabolas might intersect at zero, one, two, or even infinitely many points if they are the same parabola. However, the equations in our system represent circles, not parabolas. It's crucial to correctly identify the type of conic section represented by an equation to understand the possible solution scenarios.

Therefore, option A, which mentions parabolas and non-intersecting graphs, is incorrect in this context. Our equations represent the same circle, not parabolas, and they intersect at infinitely many points.

Concluding Thoughts

In conclusion, the system of equations:

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

has infinite solutions because the equations are dependent. They both represent the same circle, and every point on that circle satisfies both equations. Recognizing dependent equations and understanding their geometric implications is key to solving systems of equations and unraveling the mysteries of infinite solutions.

Remember, when you encounter a system of equations, always look for relationships between the equations. Are they multiples of each other? Do they represent the same geometric shape? These insights will guide you toward the solution – or in this case, the infinite solutions!

To deepen your understanding of systems of equations and their solutions, consider exploring resources like Khan Academy's Systems of Equations section. It's a fantastic place to reinforce your knowledge and tackle more challenging problems.