Solving Systems Of Equations: Infinite Solutions Explained

by Alex Johnson 59 views

In the world of mathematics, particularly when dealing with systems of equations, we often encounter scenarios where we need to find the values of variables that satisfy multiple equations simultaneously. These systems can have a unique solution, no solution, or, intriguingly, infinitely many solutions. Let's dive into the case presented, where we see an example of a system with infinitely many solutions. This article will thoroughly explain the concept, how to identify it, and what it means in practical terms.

Decoding the System of Equations

To begin, let's examine the given system of equations:

5x + 2y = 8
-4(1.25x + 0.5y = 2)

The first equation, 5x + 2y = 8, is a linear equation representing a straight line on a graph. The second equation, -4(1.25x + 0.5y = 2), looks a bit more complex initially, but we'll simplify it shortly. The core question here is: what values of x and y will satisfy both of these equations at the same time? This is where understanding the nature of the solutions becomes crucial.

Simplifying the Second Equation

The key to understanding this system lies in simplifying the second equation. Let's distribute the -4 across the terms inside the parentheses:

-4 * 1.25x = -5x
-4 * 0.5y = -2y
-4 * 2 = -8

So, the second equation becomes:

-5x - 2y = -8

Now, let's rewrite the entire system with this simplified form:

5x + 2y = 8
-5x - 2y = -8

Identifying Infinite Solutions

Here's where the magic happens. Notice anything familiar? The two equations look remarkably similar. In fact, if you multiply the second equation by -1, you get:

(-1) * (-5x - 2y) = (-1) * (-8)

Which simplifies to:

5x + 2y = 8

This is exactly the same as the first equation! This is a critical observation. When two equations in a system are essentially the same, they represent the same line on a graph. This means that any point that lies on one line will also lie on the other. Hence, there are infinitely many points (x, y) that satisfy both equations. Understanding this concept is fundamental in linear algebra.

The Elimination Method

The provided solution in the original problem demonstrates the elimination method. Let's walk through it step-by-step:

5x + 2y = 8
-4(1.25x + 0.5y = 2)

First, the second equation is multiplied by -4 (as we did before), resulting in:

5x + 2y = 8
-5x - 2y = -8

Then, the two equations are added together:

(5x + 2y) + (-5x - 2y) = 8 + (-8)

This simplifies to:

0 = 0

This result, 0 = 0, is a true statement, but it doesn't give us specific values for x and y. Instead, it confirms that the two equations are dependent, meaning they represent the same line. This is a hallmark of a system with infinitely many solutions. The implication is that we can't isolate unique values for x and y.

Why Not Other Options?

Now, let's address why the other options (A, B, and C) are incorrect:

  • A. (-4, -4): If we substitute x = -4 and y = -4 into the original equation 5x + 2y = 8, we get 5(-4) + 2(-4) = -20 - 8 = -28, which is not equal to 8. So, (-4, -4) is not a solution.
  • B. (0, 0): Substituting x = 0 and y = 0 into 5x + 2y = 8 gives us 5(0) + 2(0) = 0, which is also not equal to 8. Thus, (0, 0) is not a solution.
  • C. no solution: We've already established that the system has solutions. The equations are consistent (they don't contradict each other), and they share an infinite number of solutions.

Graphical Representation

Visualizing these equations on a graph can provide further clarity. If you were to plot both equations, you would see that they overlap perfectly. They are, in essence, the same line. This overlap visually represents the infinite number of points that satisfy both equations. This graphical approach is incredibly useful in understanding the nature of solutions.

Practical Implications of Infinite Solutions

While infinitely many solutions might seem abstract, they have practical implications in various fields. For example, in linear programming, if a system representing constraints has infinitely many solutions, it indicates a range of possible optimal solutions. In engineering, it might mean that there are multiple configurations that satisfy the design requirements.

General Strategies for Solving Systems of Equations

Understanding systems with infinite solutions is part of a broader skill set in solving systems of equations. Here are some general strategies to keep in mind:

  1. Substitution Method: Solve one equation for one variable and substitute that expression into the other equation.
  2. Elimination Method: Multiply one or both equations by constants to make the coefficients of one variable opposites, then add the equations together to eliminate that variable.
  3. Graphical Method: Plot the equations on a graph and find the point(s) of intersection. This is a powerful visual tool.
  4. Matrix Methods: For larger systems, matrix methods like Gaussian elimination or matrix inversion can be efficient.

Conclusion: The Beauty of Infinite Solutions

The solution to the system of equations presented is, therefore, D. infinitely many solutions. The key takeaway is that when simplified equations are identical or multiples of each other, they represent the same line, leading to an infinite number of solutions. Understanding this concept is crucial for mastering linear algebra and its applications. It highlights the interconnectedness of mathematical concepts and the elegance of systems that, at first glance, might seem complex.

Further explore the fascinating world of linear algebra and systems of equations by visiting trusted resources like Khan Academy's Linear Algebra section for more in-depth explanations and examples.