Solutions For Consistent And Dependent Systems Of Three Linear Equations

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In the realm of linear algebra, understanding the nature of solutions to systems of equations is paramount. A system of three linear equations with three variables can exhibit a variety of solution behaviors, broadly categorized as consistent or inconsistent, and independent or dependent. This article delves into the specific scenario of a system that is both consistent and dependent, elucidating the implications for the number of solutions it possesses.

Defining Consistency and Dependence

To effectively address the question of how many solutions exist for a consistent and dependent system, it's crucial to first define these key concepts.

  • Consistency in the context of linear systems refers to the existence of at least one solution. A consistent system is one where the equations do not contradict each other, allowing for one or more sets of variable values that satisfy all equations simultaneously.
  • Dependence, on the other hand, describes a relationship between the equations within the system. A dependent system contains at least one equation that can be derived from the others. This means that one or more equations provide redundant information, as they are essentially linear combinations of the other equations.

When a system of equations is both consistent and dependent, it indicates a unique situation where solutions exist, but the equations are not entirely independent. This interdependence directly impacts the number of solutions the system can have.

The Case of Infinitely Many Solutions

When dealing with a system of three linear equations in three variables that is both consistent and dependent, the answer to how many solutions exist is infinitely many. This stems from the fact that the dependency between equations reduces the effective number of independent constraints. Imagine you have three equations, but one of them is just a combination of the other two. In reality, you only have two independent pieces of information to define your solution set in a three-dimensional space.

To visualize this, consider each linear equation as representing a plane in three-dimensional space. For a system of three equations:

  1. If the system is consistent, it means the planes intersect. The intersection could be a single point (one unique solution), a line (infinitely many solutions), or even a plane (infinitely many solutions).
  2. If the system is dependent, at least two of the planes are either the same plane or intersect along a common line. This means there isn't a unique point of intersection; instead, the intersection is either a line or a plane.

Therefore, when both conditions are met – consistency and dependence – the planes intersect along a line or coincide as the same plane. In either case, there are infinitely many points that lie on the intersection, each representing a solution to the system of equations. This is because the third equation does not provide a new, independent constraint that would narrow the solution down to a single point.

Illustrative Examples

To solidify the understanding, let's explore a few examples of consistent and dependent systems of three linear equations.

Example 1: Planes Intersecting in a Line

Consider the following system:

x + y + z = 3
2x + 2y + 2z = 6
3x + 3y + 3z = 9

Notice that the second and third equations are simply multiples of the first equation. This indicates dependency. All three equations represent the same plane in 3D space. Any point on this plane satisfies all three equations. Thus, there are infinitely many solutions.

Example 2: Two Coinciding Planes and an Intersecting Plane

Consider the following system:

x + y + z = 3
2x + 2y + 2z = 6
x - y = 1

Here, the first two equations represent the same plane, while the third equation represents a distinct plane. The intersection of these planes will be a line. Every point on this line is a solution to the system, meaning there are infinitely many solutions.

Example 3: Expressing Solutions Parametrically

To further illustrate the concept of infinite solutions, let's solve a consistent and dependent system and express the solutions parametrically.

Consider the system:

x + y + z = 4
2x + y - z = 1
3x + 2y = 5

Solving this system (e.g., using Gaussian elimination), we find that the third equation is a linear combination of the first two. We can express the solution in terms of a parameter, say 't'. Letting z = t, we can solve for x and y in terms of t:

x = 3 - t
y = 1

Thus, the solutions are of the form (3 - t, 1, t), where t can be any real number. This parametric representation clearly shows that there are infinitely many solutions, each corresponding to a different value of t.

Contrast with Other System Types

To fully appreciate the characteristics of consistent and dependent systems, it's helpful to contrast them with other types of systems:

  • Inconsistent Systems: These systems have no solutions. Geometrically, this could mean that the planes do not intersect at all (e.g., they are parallel) or that their intersections do not coincide at any point.
  • Consistent and Independent Systems: These systems have exactly one solution. In this case, the three planes intersect at a single point, representing the unique solution.

The distinction lies in the independence of the equations. Independent equations provide unique and non-redundant information, allowing for a single solution. Dependent equations, on the other hand, introduce redundancy, leading to either infinitely many solutions (if consistent) or no solutions (if inconsistent).

Practical Implications

The concept of consistent and dependent systems extends beyond theoretical mathematics and finds applications in various fields. For example, in engineering, when designing structures, the equations representing equilibrium conditions might form a consistent and dependent system, indicating that there are multiple ways to distribute loads while maintaining stability. In economics, models with dependent equations might represent scenarios with flexibility in resource allocation.

Understanding the nature of solutions to linear systems is also crucial in numerical methods and computer simulations. When solving large systems of equations numerically, it's essential to identify dependency to avoid computational errors and to interpret the results correctly. If a system is found to be consistent and dependent, special techniques might be needed to find a representative set of solutions.

Conclusion

In summary, a system of three linear equations in three variables that is both consistent and dependent has infinitely many solutions. This is because the dependency between equations implies that at least one equation provides redundant information, and the planes represented by the equations intersect along a line or coincide as the same plane. The consistent nature ensures that there is an intersection, but the dependency prevents the intersection from being a single, unique point. By understanding the concepts of consistency and dependence, one can accurately determine the number of solutions for a system of linear equations and appreciate its implications in various practical applications.